Question

Question #753ef

Answer 1

They alternate between (local) maxes and (local) mins.

Explanation:

Let `f(x) = 3cosx`.
Then f is differentiable everywhere and
`f'(x) = -3sinx`.
`f'(x) = 0` when `sinx = 0`; that is,
for all integer multiples of `pi`.
Now find `f"(x)`.
You will discover from the Second Derivative Test that
when x is an even multiple of `pi`, `f'(x) = 0` and `f''(x) < 0`.
Therefore f is concave down at such points and f has a local maximum.
Similarly, when x is an odd multiple of `pi`, `f'(x) = 0` and `f''(x) > 0`.
Therefore f is concave up at such points and f has a local minimum.