Convexity of function. Convex direction
The concept of convexity of a function
Consider the function \(y = f\left(x \right),\) which is assumed to be continuous on the interval \(\left[ (a,b) \right].\) Function \(y = f\left(x \right )\) is called convex down (or simply convex), if for any points \((x_1)\) and \((x_2)\) from \(\left[ (a,b) \right]\) the inequality \ If this inequality is strict for any \(( x_1),(x_2) \in \left[ (a,b) \right],\) such that \((x_1) \ne (x_2),\) then the function \(f\left(x \right) \) are called strictly convex down
An upward convex function is defined similarly. The function \(f\left(x \right)\) is called convex up (or concave), if for any points \((x_1)\) and \((x_2)\) of the segment \(\left[ (a,b) \right]\) the inequality \ If this inequality is strict for any \(( x_1),(x_2) \in \left[ (a,b) \right],\) such that \((x_1) \ne (x_2),\) then the function \(f\left(x \right) \) are called strictly convex upward on the segment \(\left[ (a,b) \right].\)
Geometric interpretation of the convexity of a function
The introduced definitions of a convex function have a simple geometric interpretation.
For the function, convex down (Figure \(1\)), the midpoint \(B\) of any chord \((A_1)(A_2)\) lies higher
Similarly, for the function, convex up (Figure \(2\)), the midpoint \(B\) of any chord \((A_1)(A_2)\) lies below the corresponding point \((A_0)\) of the function graph or coincides with this point.
Convex functions have another visual property, which is related to the location tangent to the graph of the function. The function \(f\left(x \right)\) is convex down on the segment \(\left[ (a,b) \right]\) if and only if its graph lies not lower than the tangent drawn to it at any point \((x_0)\) of the segment \(\left[ (a ,b) \right]\) (Figure \(3\)).
Accordingly, the function \(f\left(x \right)\) is convex up on the segment \(\left[ (a,b) \right]\) if and only if its graph lies no higher than the tangent drawn to it at any point \((x_0)\) of the segment \(\left[ (a ,b) \right]\) (Figure \(4\)). These properties constitute a theorem and can be proven using the definition of convexity of a function.
Sufficient conditions for convexity
Let the function \(f\left(x \right)\) have its first derivative \(f"\left(x \right)\) exist on the interval \(\left[ (a,b) \right],\) and the second derivative \(f""\left(x \right)\) - on the interval \(\left((a,b) \right).\) Then the following sufficient criteria for convexity are valid:
If \(f""\left(x \right) \ge 0\) for all \(x \in \left((a,b) \right),\) then the function \(f\left(x \right )\) convex down on the segment \(\left[ (a,b) \right];\)
If \(f""\left(x \right) \le 0\) for all \(x \in \left((a,b) \right),\) then the function \(f\left(x \right )\) convex upward on the segment \(\left[ (a,b) \right].\)
Let us prove the above theorem for the case of a downward convex function. Let the function \(f\left(x \right)\) have a non-negative second derivative on the interval \(\left((a,b) \right):\) \(f""\left(x \right) \ge 0.\) Let us denote by \((x_0)\) the midpoint of the segment \(\left[ ((x_1),(x_2)) \right].\) Assume that the length of this segment is equal to \(2h.\) Then the coordinates \((x_1)\) and \((x_2)\) can be written as: \[(x_1) = (x_0) - h,\;\;(x_2) = (x_0) + h.\] Let us expand the function \(f\left(x \right)\) at the point \((x_0)\) into a Taylor series with a remainder term in Lagrange form. We get the following expressions: \[ (f\left(((x_1)) \right) = f\left(((x_0) - h) \right) ) = (f\left(((x_0)) \right) - f"\left(((x_0)) \right)h + \frac((f""\left(((\xi _1)) \right)(h^2)))((2},}
\]
\[
{f\left({{x_2}} \right) = f\left({{x_0} + h} \right) }
= {f\left({{x_0}} \right) + f"\left({{x_0}} \right)h + \frac{{f""\left({{\xi _2}} \right){h^2}}}{{2!}},}
\]
где \({x_0} - h !}
Let's add both equalities: \[ (f\left(((x_1)) \right) + f\left(((x_2)) \right) ) = (2f\left(((x_0)) \right) + \frac (((h^2)))(2)\left[ (f""\left(((\xi _1)) \right) + f""\left(((\xi _2)) \right)) \right].) \] Since \((\xi _1),(\xi _2) \in \left((a,b) \right),\) then the second derivatives on the right side are non-negative. Therefore, \ or \ that is, in accordance with the definition, the function \(f\left(x \right)\) convex down
.
Note that the necessary condition for the convexity of a function (i.e., a direct theorem in which, for example, from the condition of convexity downwards it follows that \(f""\left(x \right) \ge 0\)) is satisfied only for the non-strict inequalities. In the case of strict convexity, the necessary condition is, generally speaking, not satisfied. For example, the function \(f\left(x \right) = (x^4)\) is strictly downward convex. However, at the point \(x = 0\) its second derivative is equal to zero, i.e. the strict inequality \(f""\left(x \right) \gt 0\) does not hold in this case.
Properties of convex functions
Let us list some properties of convex functions, assuming that all functions are defined and continuous on the interval \(\left[ (a,b) \right].\)
If the functions \(f\) and \(g\) are convex downward (upward), then any of them linear combination \(af + bg,\) where \(a\), \(b\) are positive real numbers, is also convex downward (upward).
If the function \(u = g\left(x \right)\) is downward convex, and the function \(y = f\left(u \right)\) is downward convex and nondecreasing, then complex function \(y = f\left((g\left(x \right)) \right)\) will also be convex downward.
If the function \(u = g\left(x \right)\) is convex upward, and the function \(y = f\left(u \right)\) is convex downward and non-increasing, then complex function \(y = f\left((g\left(x \right)) \right)\) will be convex downward.
Local maximum upward convex function defined on the interval \(\left[ (a,b) \right],\) is also its highest value on this segment.
Local minimum downward convex function defined on the interval \(\left[ (a,b) \right],\) is also its lowest value on this segment.
Graph of a function y=f(x) called convex on the interval (a; b), if it is located below any of its tangents on this interval.
Graph of a function y=f(x) called concave on the interval (a; b), if it is located above any of its tangents on this interval.
The figure shows a curve that is convex at (a; b) and concave on (b;c).
Examples.
Let us consider a sufficient criterion that allows us to determine whether the graph of a function in a given interval will be convex or concave.
Theorem. Let y=f(x) differentiable by (a; b). If at all points of the interval (a; b) second derivative of the function y = f(x) negative, i.e. f ""(x) < 0, то график функции на этом интервале выпуклый, если же f""(x) > 0 – concave.
Proof. Let us assume for definiteness that f""(x) < 0 и докажем, что график функции будет выпуклым.
Let's take the functions on the graph y = f(x) arbitrary point M0 with abscissa x 0 Î ( a; b) and draw through the point M0 tangent. Her equation. We must show that the graph of the function on (a; b) lies below this tangent, i.e. at the same value x ordinate of curve y = f(x) will be less than the ordinate of the tangent.
So, the equation of the curve is y = f(x). Let us denote the ordinate of the tangent corresponding to the abscissa x. Then . Consequently, the difference between the ordinates of the curve and the tangent for the same value x will .
Difference f(x) – f(x 0) transform according to Lagrange's theorem, where c between x And x 0.
Thus,
We again apply Lagrange’s theorem to the expression in square brackets: , where c 1 between c 0 And x 0. According to the conditions of the theorem f ""(x) < 0. Определим знак произведения второго и третьего сомножителей.
Thus, any point on the curve lies below the tangent to the curve for all values x And x 0 Î ( a; b), which means that the curve is convex. The second part of the theorem is proved in a similar way.
Examples.
The point on the graph of a continuous function that separates its convex part from the concave part is called inflection point.
Obviously, at the inflection point, the tangent, if it exists, intersects the curve, because on one side of this point the curve lies under the tangent, and on the other side - above it.
Let us determine sufficient conditions for the fact that a given point of the curve is an inflection point.
Theorem. Let the curve be defined by the equation y = f(x). If f ""(x 0) = 0 or f ""(x 0) does not exist even when passing through the value x = x 0 derivative f ""(x) changes sign, then the point in the graph of the function with the abscissa x = x 0 there is an inflection point.
Proof. Let f ""(x) < 0 при x < x 0 And f ""(x) > 0 at x > x 0. Then at x < x 0 the curve is convex, and when x > x 0– concave. Therefore, the point A, lying on the curve, with abscissa x 0 there is an inflection point. The second case can be considered similarly, when f ""(x) > 0 at x < x 0 And f ""(x) < 0 при x > x 0.
Thus, inflection points should be sought only among those points where the second derivative vanishes or does not exist.
Examples. Find inflection points and determine the intervals of convexity and concavity of curves.
ASYMPTOTES OF THE GRAPH OF THE FUNCTION
When studying a function, it is important to establish the shape of its graph at an unlimited distance of the graph point from the origin.
Of particular interest is the case when the graph of a function, when its variable point is removed to infinity, indefinitely approaches a certain straight line.
The straight line is called asymptote function graphics y = f(x), if the distance from the variable point M graphics to this line when removing a point M to infinity tends to zero, i.e. a point on the graph of a function, as it tends to infinity, must indefinitely approach the asymptote.
A curve can approach its asymptote, remaining on one side of it or on different sides, crossing the asymptote an infinite number of times and moving from one side to the other.
If we denote by d the distance from the point M curve to the asymptote, then it is clear that d tends to zero as the point moves away M to infinity.
We will further distinguish between vertical and oblique asymptotes.
VERTICAL ASYMPTOTES
Let at x→ x 0 from any side function y = f(x) increases unlimitedly in absolute value, i.e. or or 
. Then from the definition of an asymptote it follows that the straight line x = x 0 is an asymptote. The opposite is also obvious, if the line x = x 0 is an asymptote, i.e. .
Thus, the vertical asymptote of the graph of the function y = f(x) is called a straight line if f(x)→ ∞ under at least one of the conditions x→ x 0– 0 or x → x 0 + 0, x = x 0
Therefore, to find the vertical asymptotes of the graph of the function y = f(x) need to find those values x = x 0, at which the function goes to infinity (suffers an infinite discontinuity). Then the vertical asymptote has the equation x = x 0.
Examples.
SLANT ASYMPTOTES
Since the asymptote is a straight line, then if the curve y = f(x) has an oblique asymptote, then its equation will be y = kx + b. Our task is to find the coefficients k And b.
Theorem. Straight y = kx + b serves as an oblique asymptote at x→ +∞ for the graph of the function y
= f(x) then and only when 
. A similar statement is true for x → –∞.
Proof. Let MP– length of a segment equal to the distance from the point M to asymptote. By condition . Let us denote by φ the angle of inclination of the asymptote to the axis Ox. Then from ΔMNP follows that . Since φ is a constant angle (φ ≠ π/2), then , but
To determine the convexity (concavity) of a function on a certain interval, you can use the following theorems.
Theorem 1. Let the function be defined and continuous on the interval and have a finite derivative. In order for a function to be convex (concave) in , it is necessary and sufficient that its derivative decreases (increases) on this interval.
Theorem 2. Let the function be defined and continuous along with its derivative on and have a continuous second derivative inside. For convexity (concavity) of a function in it is necessary and sufficient that inside
Let us prove Theorem 2 for the case of convex function.
Necessity. Let's take an arbitrary point. Let us expand the function around a point in a Taylor series
Equation of a tangent to a curve at a point having an abscissa:
Then the excess of the curve over the tangent to it at the point is equal to
Thus, the remainder is equal to the amount of excess of the curve over the tangent to it at point . Due to continuity, if 
, then also for , belonging to a sufficiently small neighborhood of the point , and therefore, obviously, for any value different from , belonging to the indicated neighborhood.
This means that the graph of the function lies above the tangent and the curve is convex at an arbitrary point.
Adequacy. Let the curve be convex on the interval. Let's take an arbitrary point.
Similarly to the previous one, we expand the function around a point in a Taylor series
The excess of the curve over the tangent to it at a point having an abscissa defined by the expression is equal to
Since the excess is positive for a sufficiently small neighborhood of the point, the second derivative is also positive. As we strive, we find that for an arbitrary point .
Example. Examine the function for convexity (concavity).
Its derivative 
increases on the entire number line, which means, by Theorem 1, the function is concave on .
Its second derivative 
, therefore, by Theorem 2, the function is concave on .
3.4.2.2 Inflection points
Definition. Inflection point The graph of a continuous function is the point separating the intervals in which the function is convex and concave.
From this definition it follows that the inflection points are the extremum points of the first derivative. This implies the following statements for the necessary and sufficient conditions for inflection.
Theorem (necessary condition for inflection). In order for a point to be an inflection point of a twice differentiable function, it is necessary that its second derivative at this point equals zero ( 
) or did not exist.
Theorem (sufficient condition for inflection). If the second derivative of a twice differentiable function changes sign when passing through a certain point, then there is an inflection point.
Note that at the point itself the second derivative may not exist.
The geometric interpretation of inflection points is illustrated in Fig. 3.9
In the neighborhood of a point, the function is convex and its graph lies below the tangent drawn at this point. In the neighborhood of a point, the function is concave and its graph lies above the tangent drawn at this point. At the inflection point, the tangent divides the graph of the function into convex and concave regions.
3.4.2.3 Examination of the function for convexity and the presence of inflection points
1. Find the second derivative.
2. Find the points at which the second derivative or does not exist.
Rice. 3.9.
3. Investigate the sign of the second derivative to the left and right of the found points and draw a conclusion about the intervals of convexity or concavity and the presence of inflection points.
Example. Examine the function for convexity and the presence of inflection points.
2. The second derivative is equal to zero at .
3. The second derivative changes sign at , which means the point is an inflection point.
On the interval, then the function is convex on this interval.
On the interval , which means the function is concave on this interval.
3.4.2.4 General scheme for studying functions and plotting a graph
When studying a function and plotting its graph, it is recommended to use the following scheme:
- Find the domain of definition of the function.
- Investigate the function for parity - oddness. Recall that the graph of an even function is symmetrical about the ordinate axis, and the graph of an odd function is symmetrical about the origin.
- Find vertical asymptotes.
- Investigate the behavior of a function at infinity, find horizontal or oblique asymptotes.
- Find extrema and intervals of monotonicity of the function.
- Find the intervals of convexity of the function and inflection points.
- Find the points of intersection with the coordinate axes.
The study of the function is carried out simultaneously with the construction of its graph.
Example. Explore function 
and plot it.
1. The domain of the function is .
2. The function under study is even 
, therefore its graph is symmetrical about the ordinate.
3. The denominator of the function goes to zero at , so the graph of the function has vertical asymptotes and .
The points are discontinuity points of the second kind, since the limits on the left and right at these points tend to .
4. Behavior of the function at infinity.
Therefore, the graph of the function has a horizontal asymptote.
5. Extrema and monotonicity intervals. Finding the first derivative
When , therefore, in these intervals the function decreases.
At , therefore, in these intervals the function increases.
At , therefore the point is a critical point.
Finding the second derivative
Since , then the point is the minimum point of the function.
6. Convexity intervals and inflection points.
Function at 
, which means that the function is concave on this interval.
The function for , which means that the function is convex on these intervals.
The function does not vanish anywhere, which means there are no inflection points.
7. Points of intersection with coordinate axes.
The equation has a solution, which means the point of intersection of the graph of the function with the ordinate axis (0, 1).
The equation has no solution, which means there are no points of intersection with the x-axis.
Taking into account the research carried out, it is possible to plot the function
Schematic graph of a function 
shown in Fig. 3.10.
Rice. 3.10.
3.4.2.5 Asymptotes of the graph of a function
Definition. Asymptote The graph of a function is called a straight line that has the property that the distance from point () to this straight line tends to 0 as the graph point moves indefinitely from the origin.
is differentiable on the interval (a, b). Then there is a tangent to the graph of the function at any point 
this chart ( 
), and the tangent is not parallel to the OY axis, since its angular coefficient is equal to 
, of course.
determination
on (a, b) has a release directed downwards (upwards) if it is located not below (not above) any tangent to the graph of the function on (a, b). 
(
) at all points of this interval, then the curve that is the graph of the function is convex (concave) on this interval.
is called the inflection point of the graph of a function if at the point 
the graph has a tangent, and there is such a neighborhood of the point 
, within which the graph of the function to the left and right of the point has different directions of convexity.
It is obvious that at the point of inflection the tangent intersects the graph of the function, since on one side of this point the graph lies above the tangent, and on the other - below it, i.e. in the vicinity of the point of inflection the graph of the function geometrically passes from one side of the tangent to the other and "bends" over it. This is where the name “inflection point” comes from.
continuous second derivative. Then 
.
does not have an inflection point at (0, 0), although 
at 
. Therefore, the equality of the second derivative to zero is only a necessary condition for inflection. 
has different signs to the left and right of the point, then the graph has an inflection at the point. 
has a second derivative in some neighborhood of the point, with the exception of the point itself, and there is a tangent to the graph of the function at the point 
. Then, if within the specified neighborhood it has different signs to the left and to the right of the point, then the graph of the function has an inflection at the point. 
for convexity, concavity, inflection points. 
, 
=
does not exist when 
)
)
or near points of discontinuity of the 2nd kind, it often turns out that the graph of a function approaches any given line as closely as desired. These are called straight lines.
definition 1.
is called an asymptote of a curve L if the distance from a point on the curve to this line tends to zero as the point moves away along the curve to infinity. There are three types of asymptotes: vertical, horizontal, oblique. 
is called a vertical asymptote of the graph of a function if at least one of the one-sided limits is equal to 
, i.e. or 
has a vertical asymptote 
, because 
, A 
.
If 
.
.
(
) is called the slanted asymptote of the graph of the function at 
If 
; 
.
; x=0 – vertical asymptote
.
- oblique asymptote. 
and plot it. 
function is neither even nor odd-
-
+
+
y
-4
t r.
0
Conclusion.
An important feature of the method considered is that it is based primarily on the detection and study of characteristic features in the behavior of the curve. Places where the function changes smoothly are not studied in particular detail, and there is no need for such study. But those places where the function has any peculiarities in behavior are subject to full research and the most accurate graphical representation. These features are points of maximum, minimum, points of discontinuity of the function, etc.
Determining the direction of concavity and inflections, as well as the specified method of finding asymptotes, make it possible to study functions in even more detail and obtain a more accurate idea of their graphs.
Instructions
The inflection points of a function must belong to the domain of its definition, which must be found first. The graph of a function is a line that can be continuous or have discontinuities, monotonically decrease or increase, have minimum or maximum points (asymptotes), be convex or concave. A sharp change in the last two states is called an inflection point.
A necessary condition for the existence of an inflection of a function is that the second one is equal to zero. Thus, by differentiating the function twice and equating the resulting expression to zero, we can find the abscissa of possible inflection points.
This condition follows from the definition of the properties of convexity and concavity of the graph of a function, i.e. negative and positive values of the second derivative. At the inflection point there is a sharp change in these properties, which means that the derivative passes the zero mark. However, being equal to zero is not yet enough to indicate an inflection.
There are two sufficient conditions for the fact that the abscissa found at the previous stage belongs to the inflection point: Through this point one can draw a tangent to the function. The second derivative has different signs to the right and left of the supposed inflection point. Thus, its existence at the point itself is not necessary; it is enough to determine that at it it changes sign. The second derivative of the function is equal to zero, but the third is not.
The first sufficient condition is universal and is used more often than others. Consider an illustrative example: y = (3 x + 3) ∛(x - 5).
Solution: Find the domain of definition. In this case there are no restrictions, therefore, it is the entire space of real numbers. Calculate the first derivative: y’ = 3 ∛(x - 5) + (3 x + 3)/∛(x - 5)².
Notice the appearance of the fraction. It follows from this that the domain of definition of the derivative is limited. The point x = 5 is punctured, which means that a tangent can pass through it, which partly corresponds to the first sign of sufficient inflection.
Determine the one-sided limits for the resulting expression for x → 5 – 0 and x → 5 + 0. They are -∞ and +∞. You have proven that a vertical tangent passes through the point x=5. This point may be an inflection point, but first calculate the second derivative: Y'' = 1/∛(x - 5)² + 3/∛(x - 5)² – 2/3 (3 x + 3)/∛(x - 5)^5 = (2 x – 22)/∛(x - 5)^5.
Omit the denominator since you have already taken into account the point x = 5. Solve the equation 2 x – 22 = 0. It has a single root x = 11. The last step is to confirm that the points x = 5 and x = 11 are inflection points. Analyze the behavior of the second derivative in their vicinity. Obviously, at the point x = 5 it changes sign from “+” to “-”, and at the point x = 11 - vice versa. Conclusion: both points are inflection points. The first sufficient condition is satisfied.
