MHB Finding a Min or Max point without knowing the function

[Yankel]

Hello,

I have this simple problem:

f(x) and g(x) are both differentiable. f is monotonically increasing for every x. g has a local min at x=0. we define h to be h(x)=f(g(x)).

Can we say anything about x=0 for h(x) ?

I used the chain rule to find that h'(x) = f'(g(x))*g'(x). at x=0 g'(x)=0, so h'(0) = 0 as well. is it possible to say if x=0 is min, max, or isn't is possible ?

thank you in advance.

 

[I like Serena]

We can tell that g(x) is locally increasing for x > 0. Therefore h(x) is locally increasing for x > 0 as well.
And g(x) is locally decreasing for x < 0, so h(x) is locally decreasing for x < 0 as well.
Consequently h(x) has a local minimum at x=0.
Differentiability of any of the functions is not needed.

 

[Yankel]

right, so f has no effect on h ?

 

[I like Serena]

right, so f has no effect on h ?

Well, f doesn't affect the locations of the extrema, but it does affect the value of those extrema.
And f will also affect the locations of zeroes.