State whether [tex]\( f(x) \)[/tex] has a maximum value or a minimum value, and find that value.

[tex]\[ f(x) = -5x^2 - 2x + 9 \][/tex]

Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a simplified fraction.)

A. The function has a minimum value, and the corresponding value is [tex]\(\square\)[/tex].

B. The function has a maximum value, and the corresponding value is [tex]\(\square\)[/tex].



Answer :

To determine whether the function [tex]\( f(x) = -5x^2 - 2x + 9 \)[/tex] has a maximum or minimum value, we follow these steps:

1. Find the first derivative of the function:

The first derivative [tex]\( f'(x) \)[/tex] is found by differentiating [tex]\( f(x) \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx}(-5x^2 - 2x + 9) = -10x - 2 \][/tex]

2. Find the critical points:

Critical points occur where [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -10x - 2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -10x = 2 \quad \Rightarrow \quad x = -\frac{2}{10} = -\frac{1}{5} \][/tex]

3. Determine the nature of the critical points using the second derivative:

We take the second derivative of [tex]\( f(x) \)[/tex]:
[tex]\[ f''(x) = \frac{d}{dx}(-10x - 2) = -10 \][/tex]

Evaluate the second derivative at the critical point [tex]\( x = -\frac{1}{5} \)[/tex]:
[tex]\[ f''\left( -\frac{1}{5} \right) = -10 \][/tex]

Since [tex]\( f''\left( -\frac{1}{5} \right) < 0 \)[/tex], the concavity at [tex]\( x = -\frac{1}{5} \)[/tex] is downward, indicating a local maximum at this critical point.

4. Find the value of the function at the critical point:

Substitute [tex]\( x = -\frac{1}{5} \)[/tex] back into the original function [tex]\( f(x) \)[/tex] to find the corresponding value:
[tex]\[ f\left( -\frac{1}{5} \right) = -5 \left( -\frac{1}{5} \right)^2 - 2 \left( -\frac{1}{5} \right) + 9 \][/tex]
Simplify the expression:
[tex]\[ = -5 \left( \frac{1}{25} \right) + \frac{2}{5} + 9 \][/tex]
[tex]\[ = -\frac{5}{25} + \frac{2}{5} + 9 \][/tex]
[tex]\[ = -\frac{1}{5} + \frac{2}{5} + 9 \][/tex]
[tex]\[ = \frac{1}{5} + 9 \][/tex]
[tex]\[ = \frac{1}{5} + \frac{45}{5} \][/tex]
[tex]\[ = \frac{46}{5} \][/tex]

Therefore, the function [tex]\( f(x) \)[/tex] has a maximum value and the corresponding value is [tex]\( \frac{46}{5} \)[/tex]. So, the correct choice is:

B. The function has a maximum value and the corresponding value is [tex]\(\frac{46}{5}\)[/tex].