Answered

Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

If g is a differentiable function such that g(x)<0 for all real numbers x and if f'(x) = (x2 - 4)g(x), which of the following is true? i. f has relative maxima at x-2 and at x-2.
ii. f has relative minima at x=-2 and at x=2.
iii. It cannot be determined if fhas any relative extrema.
iv. f has a relative maximum at x=-2 and a relative minimum at x=2.
v. f has a relative minimum at x=-2 and a relative maximum at x=2.

Sagot :

facundo3141592

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.

Maximums and minimums.

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:

https://brainly.com/question/1938915

Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.