Question

Suppose f(x) is a function which satisfies f'(3) = 0,f'(5) = 0, f"(3) = -4, and f"(5) = 5.
Which of the following statements is true?
A. f(x) has a relative minimum at x = 3 and at x = 5.
B. f(x) has a relative minimum at x = 3 and a relative maximum at x = 5.
C. f(x) has a relative maximum at x = 3 and at x = 5.
D. f(x) has a relative maximum at x = 3 and a relative minimum at x = 5.
E. None of the above is true,


Any help​

Answer 1

if you find the answer let me know please e

Answer 2

Using the concept of critical point and the second derivative test, it is found that the correct option is:

D. f(x) has a relative maximum at x = 3 and a relative minimum at x = 5.

• The critical points of a function $f(x)$ are the values of x for which $f^{\prime}(x) = 0$.

Applying the second derivative test at the critical points, we have that:

• If positive, that is, $f^{\prime\prime}(x) > 0$, it is a relative minimum.

• If negative, that is, $f^{\prime\prime}(x) < 0$, it is a relative maximum.

• If zero, that is, $f^{\prime\prime}(x) = 0$, we do not have sufficient information.

In this problem:

• $f^{\prime}(3) = 0, f^{\prime\prime}(3) < 0$, thus, there is a maximum at x = 3.

• $f^{\prime}(5) = 0, f^{\prime\prime}(5) > 0$, thus, there is a minimum at x = 5.

The correct option is:

D. f(x) has a relative maximum at x = 3 and a relative minimum at x = 5.

A similar problem is given at https://brainly.com/question/16944025