Question

Suppose you know that is a differentiable function and the domain is all real numbers, and that f' > 0 for x on the interval [-a,0), f' = 0 at x = 0 and f’ < 0 for x on the interval (0,a] for some positive constant a.Then you know that f has a ___ at x=0

Answer 1

Based on the given information, if f' > 0 for x on the interval [-a,0), f' = 0 at x = 0, and f' < 0 for x on the interval (0,a], then f has a local maximum at x = 0.

Since f' > 0 for x on the interval [-a,0), this indicates that the function f is increasing in that interval. At x = 0, where f' = 0, the function reaches a critical point. Then, for x on the interval (0,a], where f' < 0, the function f is decreasing.

Considering the behaviour of f on both sides of x = 0, we observe that the function transitions from increasing to decreasing. This implies that the function reaches a local maximum at x = 0.

It's important to note that this conclusion is based on the given conditions and assumptions about the behaviour of f' (the derivative of f) on the intervals specified.