1. Calculus Help. ANy OnE?

Question

If, for all x, f ' (x) = (x - 2)^4 (x-1)^3, it follows that the function f has

A-a relative min @ x=1
B-a relative max @ x=1
C-both a relative min @ x=1 and a relative max @ x=2
D. neither a relative min nor a relative max.
E-relaztive minima @ x=1 and at x=2

Answer 1

The sign of f'(x) is---------1++++++++2++++++++++
so at x=1 there is a relative minimum
at x=2 the derivative does not change sign so at x=2 there is NO maximum nor minimum
The answer is A

Answer 2

Consider that (x-2)^4 will always be non-negative. On the other hand, (x-1)^3 can be positive or negative depending on the value of x.

For x < 1, x - 1 < 0 and (x-1)^3 < 0, so f'(x) < 0. Therefore, f(x) must be decreasing on the interval [-inf, 1).

For x > 1, (x-1) > 0 and (x-1)^3 > 0, so f'(x) > 0.. Therefore, f(x) must be increasing on the interval (1, +inf].

So there must be a relative minimum at x=1, but no relative extremum at x=2 (since f(x) is increasing over (1,2) as well as over (2, +inf] ).