Find

a) any critical values and

b) any relative extrema for the function

[tex]\[ g(x) = -x^3 + 12x - 7 \][/tex]

a) Select the correct choice below and, if necessary, fill in the answer box within your choice.

A. The critical value(s) of the function is/are [tex]$\square$[/tex] [tex]$\square$[/tex].
(Use a comma to separate answers as needed.)

B. The function has no critical values.



Answer :

To solve the problem, let's break it down step by step.

### a) Finding the Critical Values

1. Start with the original function:
[tex]\[ g(x) = -x^3 + 12x - 7 \][/tex]

2. Find the first derivative [tex]\( g'(x) \)[/tex]:
[tex]\[ g'(x) = \frac{d}{dx} (-x^3 + 12x - 7) = -3x^2 + 12 \][/tex]

3. Set the first derivative equal to zero to find critical values:
[tex]\[ -3x^2 + 12 = 0 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ -3x^2 + 12 = 0 \implies -3x^2 = -12 \implies x^2 = 4 \implies x = \pm 2 \][/tex]

So, the critical values are [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex].

### b) Determining the Nature of Each Critical Point (Relative Extrema)

1. Find the second derivative [tex]\( g''(x) \)[/tex]:
[tex]\[ g''(x) = \frac{d}{dx} (-3x^2 + 12) = -6x \][/tex]

2. Analyze the second derivative at the critical values:

- For [tex]\( x = -2 \)[/tex]:
[tex]\[ g''(-2) = -6(-2) = 12 \quad (\text{positive, indicating a relative minimum}) \][/tex]

- For [tex]\( x = 2 \)[/tex]:
[tex]\[ g''(2) = -6(2) = -12 \quad (\text{negative, indicating a relative maximum}) \][/tex]

### Summary

- Critical values:
[tex]\[ x = -2, 2 \][/tex]

- Relative extrema:
- [tex]\( x = -2 \)[/tex] is a relative minimum.
- [tex]\( x = 2 \)[/tex] is a relative maximum.

### Answer:
a) The critical values are [tex]\( -2 \)[/tex] and [tex]\( 2 \)[/tex].
[tex]\[ A. The critical value(s) of the function is/are \boxed{-2, 2}. \][/tex]

b) The relative extrema are:
- [tex]\( x = -2 \)[/tex] is a relative minimum.
- [tex]\( x = 2 \)[/tex] is a relative maximum.