Calculus and Differentiation

New to this forum

Have a question from the Spivak’s Calculus
Show the relationship between the critical points of f and those from f^2

My thought is to divide into 2 cases where f'(x) equals 0 or f'(x) does not exist

Then the first case is quite easy to prove simply by expressing (f^2(x))’ into 2f(x)f'(x) since f(x) exists for f'(x) to stand then we can conclude that (f^2(x))’ equals to 0

But it gets troublesome when it comes to the case where f is not differentiable at x since (f^2(x))’may equal to 0 or not exist

I also come up some example:
f(x)=|x|–(f^2(0))’=0 f'(0) does not exist
f(x)=1/x — (f^2(0))’ and f'(0) both do not exist

Do you have a idea on how I should continue?




Use to put your maths text in i.e with f(x) inside it gives f(x)f(x)f(x)
– Mattos
2 days ago