if $f(a) = f(b) = 0$ and $f(c)\neq0,$ then there exist $x_1 , x_2 \in \mathbb R $ s.t $f'(x_1) > 0$ and$ f'(x_2) <0$

Question

Given that $f$ is diff on $(a,b)$ and cts on $[a,b]$ and $c\in (a,b)$

My approach: Case $1$: $f(c) > 0$,

Take the interval $(a,c)$

By MVT, there exist $x_1$ s.t ,

$f(c) - f(a) = f'(x_1) (c-a)$ which is $f(c) = f'(x_1)(c-a) > 0 \rightarrow f'(x_1) > 0$

Now take the interval $(c,b)$

By MVT, there exist $x_2$ s.t

$f(b) - f(c) = f'(x_2) (b-c)$ which is $-f(c) = f'(x_2)(b-c) < 0 \rightarrow f'(x_2) < 0$

Same goes for $f(c) <0 $

am I all good here?

Answer 1

Due to Rolle's theorem there exists $x=c$ s. t. $f'(c)=0$ indicating max/min at $x=c$, then for some $x$ value $f'(x)$ will be positive and for some other value it will be negative.