We want to see what can we say about f(x) extremes for the given information. We will see that the correct option is:
V: f has a relative minimum at x=-2 and a relative maximum at x=2.
<h3>Maximums and minimums.</h3>
A given function f(x) will have a maximum/minimum at x₀ if:
f'(x₀) = 0
- It will be a maximum if f''(x₀) < 0
- It will be a minimum if f''(x₀) > 0.
Here we do know that:
- f'(x) = (x^2 - 4)*g(x)
- g(x) < 0 for all numbers x.
Then the only zeros of f'(x) are when (x^2 - 4) = 0.
And this happens for x = 2 and x = -2
Now let's see if these are minimums or maximums, we have:
f''(x) = 2*x*g(x) + (x^2 - 4)*g'(x)
If we evaluate this in x = 2, we get:
f''(2) = 2*2*g(2) + (2^2 - 4)*g'(2)
= 4*g(2) + 0*g'(2) = 4*g(2)
And g(x) is always negative, then 4*g(2) < 0, then:
f''(2) < 0
Meaning that in x = 2 we have a relative maximum.
For x = -2 we have:
f''(-2) = 2*-2*g(-2) + ((-2)^2 - 4)*g'(-2)
= -4*g(2) > 0
Then f''(-2) > 0, meaning that in x = -2 we have a relative minimum.
Then the correct option is:
V: f has a relative minimum at x=-2 and a relative maximum at x=2.
If you want to learn more about maximums and minimums, you can read:
brainly.com/question/1938915
