Is the function still monotonic?

Question

Let $f:[a,b]\to \mathbb{R}$ be a function which is continuous on $[a,b]$ and differentiable on $(a,b)$. If $\exists c\in (a,b)$ such that $f'(c)=0$,$f'(x)<0, \forall x\in (a,c)$ and $f'(x)>0,\forall x\in (c,a)$, then is it true that $f$ is strictly decreasing on $[a,c]$ and strictly increasing on $[c,b]$?
My thoughts: $f$ is definitely strictly decreasing on $(a,c]$ and strictly increasing on $[c,b)$. I think that due to continuity the desired conclusion should follow, but I am not sure because I am unable to prove this.
I came up with this while studying the local extrema of $f:[-1,1]\to \mathbb{R}$,$f(x)=\arcsin x^2$. This function satisfies all the hypothesis I mentioned above, and the required conclusion is true ( I could tell this by looking at the graph of the function), but I can't prove it rigorously.

Answer 1

Could it happen that $f|_{[c,b]}$ is not strictly increasing? Since you know that $f|_{(c,b]}$ is strictly increasing, this would mean that there was some $d\in(c,b]$ such that $f(d)<f(c)$. But then $\frac{f(d)-f(c)}{d-c}<0$, which is impossible, since $\frac{f(d)-f(c)}{d-c}=f'(x_0)$ for some $x_0\in(c,d)$ and $f'(x_0)>0$.