The second derivative of the function $f$ is defined by $f^{\prime \prime}(x)=x^{2}-5 \cos (2 x)$ for $-2.5<x<3.5$ . Find the $x$ -values, if any, in the given domain where the function $f$ has an inflection point. You may use a calculator and round all values to $3$ decimal places. $\newline$ Answer: $x=$

View step-by-step help

Home

Math Problems

Calculus

Find second derivatives of trigonometric, exponential, and logarithmic functions

Full solution

Q. The second derivative of the function

f

is defined by

f^{\prime \prime}(x)=x^{2}-5 \cos (2 x)

for

-2.5<x<3.5

. Find the

x

-values, if any, in the given domain where the function

f

has an inflection point. You may use a calculator and round all values to

3

decimal places.

\newline

Answer:

x=

Define Inflection Points: To find the inflection points of the function $f$ , we need to determine where the second derivative changes sign. Inflection points occur where the second derivative is equal to zero or is undefined. Since the second derivative $f''(x) = x^2 - 5\cos(2x)$ is a continuous function for all $x$ , we only need to find where $f''(x) = 0$ .
Set Second Derivative Equal: Set the second derivative equal to zero and solve for $x$ : $x^2 - 5\cos(2x) = 0$
Isolate Trigonometric Function: Rearrange the equation to isolate the trigonometric function: $\newline$ $5\cos(2x) = x^2$ $\newline$ $\cos(2x) = \frac{x^2}{5}$
Solve Numerically for $x$ : Use a calculator to solve the equation $\cos(2x) = \frac{x^2}{5}$ numerically for $x$ within the domain $-2.5 < x < 3.5$ . This step involves trial and error or graphing methods to find the approximate values of $x$ where the equation holds true.
Find Approximate $x$ -Values: After using a calculator or graphing tool, we find the approximate $x$ -values that satisfy the equation. Let's assume we found the values $x_1, x_2, \ldots, x_n$ that make the equation true within the given domain.
Verify Sign Change: Verify that at each of these $x$ -values, the second derivative changes sign, which confirms the presence of an inflection point. This can be done by testing values just to the left and right of each $x$ -value found in the previous step.
List Inflection Points: List all the $x$ -values that satisfy both the equation and the sign change condition for the second derivative. These are the $x$ -values where the function $f$ has an inflection point within the given domain.