Let f and g be twice differentiable functions such that f'(x) ≤ 0 for all x in the domain of f.

Question

If h(x)=f(g'(x)) and h'(2)=4, what (if anything) can you say about f, g, and h at x=2?

Answer 1

f'(x) ≤ 0 means f(x) is a decreasing function for all x. Noe let's take derivative of  h(x)=f(g'(x))

h'(x)=f'(g'(x))g"(x) 

f'(g'(x))≤ 0.

Since f'(g'(x))≤ 0,  h'(2)=4,  f'(g'(2))g"(2)=2, therefore g''(2)≤ 0.

f'(g'(2))g''(2))=2.

At x=2

 h(x) is increasing

 f(x)  is decreasing

 g(x) is desreasing

You can make more conclusions if you need.