Problem

Suppose that $f$ is a differentiable function on $(0,1)$ and $f$ is continuous on $[0,1]$. Which of the following statement is false?

(a) If $f(0)f(1)<0$, then there exists a number $c\in(0,1)$ such that $f(c)=0$.

(b) If $f'(0.1)f'(0.9)<0$, then there exists a number $c\in(0.1,0.9)$ such that $f'(c)=0$.

(c) If $f(0)=0$ and $f(1)=1$, then there exists a number $c\in(0,1)$ such that $f'(c)=1$.

(d) If $f''>0$ on $(0,1)$ and $f'(c)=0$ for some $c\in(0,1)$, then $f(x)\ge f(c)$ for $x\in[0,1]$.

(e) If $f(x)=f(1-x)$ for $x\in[0,1]$, $f(0)>0$, and $f(0.5)=0$, then $f''(0.5)>0$.