Question

find the absolute extrema if they exist, as well as all values of x where they occur, for the function f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-10x + 1 on the domain -3,6.
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum is \square, which occurs at x = \square.
(round the absolute maximum to two decimal places as needed. type an exact answer for the value of x where the maximum occurs. use a comma to separate answers as needed.)
b. there is no absolute maximum.
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is \square, which occurs at x = \square.
(round the absolute minimum to two decimal places as needed. type an exact answer for the value of x where the minimum occurs. use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

Step1: Find the derivative of the function

The function is \(f(x)=\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-10x + 1\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(f^\prime(x)=x^{2}-3x - 10\).

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(x^{2}-3x - 10 = 0\).
Factor the quadratic equation: \((x - 5)(x+ 2)=0\).
Solving \((x - 5)(x + 2)=0\) gives \(x = 5\) or \(x=-2\).

Step3: Evaluate the function at critical points and endpoints

• For \(x=-3\): \(f(-3)=\frac{1}{3}(-3)^{3}-\frac{3}{2}(-3)^{2}-10(-3)+1=-9-\frac{27}{2}+30 + 1=\frac{-18-27 + 60+2}{2}=\frac{17}{2}=8.5\).
• For \(x=-2\): \(f(-2)=\frac{1}{3}(-2)^{3}-\frac{3}{2}(-2)^{2}-10(-2)+1=-\frac{8}{3}-6 + 20+1=\frac{-8-18 + 60 + 3}{3}=\frac{37}{3}\approx12.33\).
• For \(x = 5\): \(f(5)=\frac{1}{3}(5)^{3}-\frac{3}{2}(5)^{2}-10(5)+1=\frac{125}{3}-\frac{75}{2}-50 + 1=\frac{250-225-300 + 6}{6}=-\frac{269}{6}\approx-44.83\).
• For \(x = 6\): \(f(6)=\frac{1}{3}(6)^{3}-\frac{3}{2}(6)^{2}-10(6)+1=72-54-60 + 1=-41\).

Answer:

A. The absolute maximum is \(12.33\), which occurs at \(x=-2\).
A. The absolute minimum is \(-44.83\), which occurs at \(x = 5\).