Question

1. Find the relative extrema, if any, of . f(t) = t - 7lnt, Use the Second Derivative Test, if possible.

a.
relative minimum: none, relative maximum: f(1/7) = 1/7 + 7ln7
b.
relative minimum: none, relative maximum: f(7) = 7 - 7ln7
c.
relative minimum: f(1/7) = 1/7 + 7ln7, relative maximum: none
d.
relative minimum: f(7) = 7-7ln7, relative maximum: none

Answer 1

Explanation:

The relative extrema can be found by finding the critical points and using the second derivative test.

To find the critical points, we need to find where the derivative of f(t) equals zero or is undefined.

f'(t) = 1 - 7/t

Setting f'(t) = 0, we have:

1 - 7/t = 0

Solving for t, we get:

t = 7

Since f'(t) is defined for all values of t, there are no other critical points.

To use the second derivative test, we need to find the second derivative of f(t).

f''(t) = 7/t^2

For t = 7, f''(7) = 7/7^2 = 1/7

Since f''(7) is positive, we can conclude that f(t) has a relative minimum at t = 7.

Therefore, the answer is:

b. relative minimum: none, relative maximum: f(7) = 7 - 7ln7