Question

the graph and equation of the function f are given.
a. use the graph to find any values at which f has a relative maximum, and use
the equation to calculate the relative maximum for each value.
b. use the graph to find any values at which f has a relative minimum, and use
the equation to calculate the relative minimum for each value.
$f(x) = 2x^3 - 3x^2 - 12x + 8$
graph with window $-5.5,1$ by $-18,18.2$

a. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
○ a. the function f has (a) relative maxima(maximum) at \\(\square\\) and the relative maxima(maximum) are(is) \\(\square\\).
(use a comma to separate answers as needed.)
○ b. the function f has no relative maxima.

Explanation:

Step1: Find critical points via derivative

To find relative maxima/minima, first find the derivative of \( f(x) = 2x^3 - 3x^2 - 12x + 8 \). The derivative \( f'(x) = 6x^2 - 6x - 12 \). Set \( f'(x) = 0 \):
\( 6x^2 - 6x - 12 = 0 \)
Divide by 6: \( x^2 - x - 2 = 0 \)
Factor: \( (x - 2)(x + 1) = 0 \)
So critical points at \( x = -1, x = 2 \).

Step2: Analyze critical points (or use graph)

From the graph (or second derivative test), \( x = -1 \) is a relative maximum (since the graph rises then falls here). Plug \( x = -1 \) into \( f(x) \):
\( f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 8 = -2 - 3 + 12 + 8 = 15 \).

Answer:

A. The function \( f \) has a relative maximum at \( -1 \) and the relative maximum is \( 15 \).