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# Calculus

Applications of the Derivative

# Inflection Points

Page 1
Problems 1-2
Page 2
Problems 3-10

### Definition of an Inflection Point

Consider a function $$y = f\left( x \right),$$ which is continuous at a point $${x_0}.$$ The function $$f\left( x \right)$$ can have a finite or infinite derivative $$f’\left( {{x_0}} \right)$$ at this point. If, when passing through $${x_0}$$, the function changes the direction of convexity, i.e. there exists a number $$\delta \gt 0$$ such that the function is convex upward on one of the intervals $$\left( {{x_0} – \delta ,{x_0}} \right)$$ or $$\left( {{x_0},{x_0} + \delta } \right)$$, and is convex downward on the other, then $${x_0}$$ is called a point of inflection of the function $$y = f\left( x \right).$$

The geometric meaning of an inflection point is that the graph of the function $$f\left( x \right)$$ passes from one side of the tangent line to the other at this point, i.e. the curve and the tangent line intersect (see Figure $$1$$).

Another interesting feature of an inflection point is that the graph of the function $$f\left( x \right)$$ in the vicinity of the inflection point $${x_0}$$ is located within a pair of the vertical angles formed by the tangent and normal (Figure $$2$$).

Figure 1.

Figure 2.

### Necessary Condition for an Inflection Point (Second Derivative Test)

If $${x_0}$$ is a point of inflection of the function $$f\left( x \right)$$, and this function has a second derivative in some neighborhood of $${x_0},$$ which is continuous at the point $${x_0}$$ itself, then

$f^{\prime\prime}\left( {{x_0}} \right) = 0.$

Proof.
Suppose that the second derivative at the inflection point $${x_0}$$ is not zero: $$f^{\prime\prime}\left( {{x_0}} \right) \ne 0.$$ Since it is continuous at $${x_0},$$ then there exists a $$\delta$$-neighborhood of the point $${x_0}$$ where the second derivative preserves its sign, i.e.

${f^{\prime\prime}\left( {{x_0}} \right) \lt 0\;\;\text{or}\;\;\;}\kern-0.3pt{f^{\prime\prime}\left( {{x_0}} \right) \lt 0\;\forall \;x \in \left( {{x_0} – \delta ,{x_0} + \delta } \right).}$

In this case, the function is either strictly convex upward (when $$f^{\prime\prime}\left( x \right) \lt 0$$) or strictly convex downward (when $$f^{\prime\prime}\left( x \right) \gt 0$$). But then the point $${x_0}$$ is not an inflection point. Hence, the assumption is wrong and the second derivative of the inflection point must be equal to zero.

### First Sufficient Condition for an Inflection Point (Second Derivative Test)

If the function $$f\left( x \right)$$ is continuous and differentiable at a point $${x_0},$$ has a second derivative $$f^{\prime\prime}\left( {{x_0}} \right)$$ in some deleted $$\delta$$-neighborhood of the point $${x_0}$$ and if the second derivative changes when passing through the point $${x_0},$$ then $${x_0}$$ is a point of inflection of the function $$f\left( x \right).$$

Proof.
Suppose, for example, that the second derivative $$f^{\prime\prime}\left( x \right)$$ changes sign from plus to minus when passing through the point $${x_0}.$$ Hence, in the left $$\delta$$-neighborhood $$\left( {{x_0} – \delta ,{x_0}} \right),$$ the inequality $$f^{\prime\prime}\left( x \right) \gt 0,$$ holds, and in the right $$\delta$$-neighborhood $$\left( {{x_0},{x_0} + \delta } \right),$$ the inequality $$f^{\prime\prime}\left( x \right) \lt 0$$ is valid.

In this case, according to the sufficient conditions for convexity, the function $$f\left( x \right)$$ is convex downward in the left $$\delta$$-neighborhood of the point $${x_0}$$ and is convex upward in the right $$\delta$$-neighborhood.

Consequently, the function changes the direction of convexity at the point $${x_0},$$ i.e. by definition, $${x_0}$$ is a point of inflection.

### Second Sufficient Condition for an Inflection Point (Third Derivative Test)

Let $$f^{\prime\prime}\left( {{x_0}} \right) = 0,$$ $$f^{\prime\prime\prime}\left( {{x_0}} \right) \ne 0.$$ Then $${x_0}$$ is a point of inflection of the function $$f\left( x \right).$$

Proof.
As $$f^{\prime\prime\prime}\left( {{x_0}} \right) \ne 0,$$ the second derivative is either strictly increasing at $${x_0}$$ (if $$f^{\prime\prime\prime}\left( {{x_0}} \right) \gt 0$$) or strictly decreasing at this point (if $$f^{\prime\prime\prime}\left( {{x_0}} \right) \lt 0$$). Because $$f^{\prime\prime}\left( {{x_0}} \right) = 0,$$ then the second derivative for some $$\delta \gt 0$$ has different signs in the left and right $$\delta$$-neighborhood of $${x_0}.$$ Hence, on the basis of the previous theorem, it follows that $${x_0}$$ is a point of inflection of the function $$f\left( x \right).$$

## Solved Problems

Click on problem description to see solution.

### ✓Example 1

Determine whether the point $$x = 0$$ is an inflection point of the function $$f\left( x \right) = \sin x.$$

### ✓Example 2

Find the points of inflection of the function

${f\left( x \right) \text{ = }}\kern0pt{ {x^4} – 12{x^3} + 48{x^2} + 12x + 1.}$

### ✓Example 3

Find the points of inflection of the function $$f\left( x \right) = {x^2}\ln x.$$

### ✓Example 4

Find the points of inflection of the function $$f\left( x \right) = {e^{ – {x^2}}}.$$

### ✓Example 5

Find the points of inflection of the function $$f\left( x \right) = {e^{\sin x}}.$$

### ✓Example 6

For what values of $$a$$ and $$b$$ the point $$\left( { – 1,2} \right)$$ is an inflection point of the graph of the function $$y\left( x \right) = a{x^3} + b{x^2}?$$

### ✓Example 7

Find the inflection points of a Gaussian function.

### ✓Example 8

Find the points of inflection of the function

$f\left( x \right) = \sqrt[\large 3\normalsize]{{{x^2}\left( {x + 1} \right)}}.$

### ✓Example 9

Find the inflection points of the curve defined by the parametric equations:

$x = {t^2},\;\;y = t + {t^3}.$

### ✓Example 10

Show that the graph of the function $$y = {\large\frac{{x + 1}}{{{x^2} + 1}}\normalsize}$$ has three points of inflection lying on one straight line.

### Example 1.

Determine whether the point $$x = 0$$ is an inflection point of the function $$f\left( x \right) = \sin x.$$

#### Solution.

We use the second sufficient condition for the existence of an inflection point.
Find the derivatives of the sine function up to the third order:

${f’\left( x \right) = {\left( {\sin x} \right)^\prime } = \cos x,\;\;\;}\kern-0.3pt {f^{\prime\prime}\left( x \right) = {\left( {\cos x} \right)^\prime } = – \sin x,\;\;\;}\kern-0.3pt {f^{\prime\prime\prime}\left( x \right) = {\left( { – \sin x} \right)^\prime } = – \cos x.}$

At the point $$x = 0,$$ the second and third derivatives have the following values:

${f^{\prime\prime}\left( 0 \right) = – \sin 0 = 0,\;\;\;}\kern-0.3pt {f^{\prime\prime\prime}\left( 0 \right) = – \cos 0 = – 1.}$

Thus, we have $$f^{\prime\prime}\left( {{x_0}} \right) = 0,$$ $$f^{\prime\prime\prime}\left( {{x_0}} \right) \ne 0.$$ Hence, by the second sufficient condition, the point $$x = 0$$ is a point of inflection.

### Example 2.

Find the points of inflection of the function

${f\left( x \right) \text{ = }}\kern0pt{ {x^4} – 12{x^3} + 48{x^2} + 12x + 1.}$

#### Solution.

Find the derivatives:

${f’\left( x \right) } = {{\left( {{x^4} – 12{x^3} + 48{x^2} + 12x + 1} \right)^\prime } } = {4{x^3} – 36{x^2} + 96x + 12 } = {4\left( {{x^3} – 9{x^2} + 24x + 3} \right);}$
${f^{\prime\prime}\left( x \right) } = {{\left( {4\left( {{x^3} – 9{x^2} + 24x + 3} \right)} \right)^\prime } } = {4\left( {3{x^2} – 18x + 24} \right) } = {12\left( {{x^2} – 6x + 8} \right).}$

Calculate the roots of the second derivative:

${f^{\prime\prime}\left( x \right) = 0,\;\;}\Rightarrow {12\left( {{x^2} – 6x + 8} \right) = 0,\;\;}\Rightarrow {{x^2} – 6x + 8 = 0,\;\;}\Rightarrow {{x_1} = 2,\;{x_2} = 4.}$

In this case it is convenient to use the second sufficient condition for the existence of an inflection point. The third derivative is written as

${f^{\prime\prime\prime}\left( x \right) } = {{\left( {12\left( {{x^2} – 6x + 8} \right)} \right)^\prime } } = {12\left( {2x – 6} \right) = 24\left( {x – 3} \right).}$

From this we immediately see that the third derivative is not zero at the points $${x_1} = 2$$ and $${x_2} = 4.$$ Therefore, these points are points of inflection.

Page 1
Problems 1-2
Page 2
Problems 3-10