Question

True or false:

Answer 1

1. False. $f'(c)=0$ does not necessarily mean that $f(c)$ is a maximum or minimum. It could just as easily be a saddle point. For example, consider the function $f(x)=x^3$ with $f'(x)=3x^2=0$ when $x=0$, yet $f(x)$ for $x$ and $f(x)>0$ for $x>0$.

2. True. If $f'(x)=0$ for all $x$ in $(a,b)$, then $f$ must be constant on that interval.

3. False. In order for the MVT to apply, $f$ must be continuous on the closed interval. But $\dfrac1{x^2}$ does not exist at $x=0$.