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The derivative of the function
f is defined by
f^(')(x)=x^(2)-5-4sin(2x). What is the
x-coordinate of the absolute maximum value of the function
f on the closed interval
[-4,3] ? You may use a calculator and round your answer to the nearest thousandth.

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-5-4 \sin (2 x)$ . What is the $x$ -coordinate of the absolute maximum value of the function $f$ on the closed interval $[-4,3]$ ? You may use a calculator and round your answer to the nearest thousandth.

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Domain and range of square root functions: equations

Full solution

Q. The derivative of the function

f

is defined by

f^{\prime}(x)=x^{2}-5-4 \sin (2 x)

. What is the

x

-coordinate of the absolute maximum value of the function

f

on the closed interval

[-4,3]

? You may use a calculator and round your answer to the nearest thousandth.

Identify Critical Points: Identify the critical points of the function $f$ by finding the values of $x$ where the derivative $f'(x)$ is equal to zero or undefined. $\newline$ $f'(x) = x^2 - 5 - 4\sin(2x)$ $\newline$ Set the derivative equal to zero to find the critical points: $\newline$ $x^2 - 5 - 4\sin(2x) = 0$ $\newline$ This equation may require numerical methods or a calculator to solve.
Find Critical Points: Use a calculator to find the approximate values of $x$ that satisfy the equation from Step $1$ . $\newline$ After using a calculator, we find that the critical points within the interval $[-4,3]$ are approximately $x = -2.279$ , $x = 0.632$ , and $x = 2.546$ . (Values are rounded to the nearest thousandth.)
Evaluate Function: Evaluate the function $f$ at the critical points and at the endpoints of the interval to determine the absolute maximum value. $\newline$ We need to evaluate $f$ at $x = -4$ , $x = -2.279$ , $x = 0.632$ , $x = 2.546$ , and $x = 3$ . $\newline$ Since we do not have the original function $f$ , we cannot directly evaluate $f$ at these points. We need the antiderivative of $f'(x)$ to find $f$ $0$ . Without the original function $f$ , we cannot complete this step.

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