A continuous function $f$ on $[a,b]$ A continuous function $f$ on $[a,b]$, differentiable in $(a,b)$, has only 1 point where its derivative vanishes. What is true about this function?
A. $f$ cannot have an even number of extrema.
B. $f$ cannot have a maximum at one endpoint and minimum at the other.
C. $f$ might be monotonically increasing.
I think $A$ is true, since it has three extrema, but the right answer should be $C$, why?
 A: To see that A is false, consider the function $f(x) = (x-0.5)^2$ on [-1,1].  To see that B is false and also that C is true, consider the function $f(x) = x^3$ on the same domain.
A: If $f$ is differentiable on $[a,b]$ then the example below shows only C works

$\bbox[5px,border:2px solid]{f(x)=-x^2\text{ on interval }[-1,0]}$ with vanishing point $f'(0)=0$


*

*A. false, $f(-1)$ minimum and $f(0)$ maximum, thus $2$ extrema

*B. false, for the same reason

*C. true, $f$ is $\nearrow$ 


Since the might is used for C, this is the only possible answer (there exists some $f$ that verify C).

But: you stated $f$ only differentiable on $]a,b[$
So if we are not in the previous case then we have to consider vertical tangents either in $a^+$ or $b-$ or both.
We shall see that even in this case, we can find functions such that only C is valid.

$\bbox[5px,border:2px solid]{f(x)=(1-x)\sqrt{x}\text{ on interval }[0,\frac 13]}$ with vanishing point $f'(\frac 13)=0$
Is not differentiable in $0$, has a minimum in $f(0)$ and a maximum in $f(\frac 13)$ and $f\ \nearrow$.
So again A. false, B. false and C. true
I let you convince yourself that the situation is the same if we exchange the role of $a$ and $b$.

Now what about the case where $f'$ is not defined nor in $a$ nor in $b$.

$\bbox[5px,border:2px solid]{f(x)=\sqrt{x}-\sqrt{2-x}-x\text{ on interval }[0,2]}$ with vanishing point $f'(1)=0$
Is not differentiable in $x=0\text{ or }2$, has minimum $f(0)$, maximum $f(2)$ and $f(1)$ is an inflexion point not an extrema, finally $f\ \nearrow$.
So again A. false, B. false and C. true
A: I argue that the truthfulness of A depends on if “extrema” means “relative extrema,” “global extrema,” or both. After all, a function has to have a minimum and maximum value on an internal. For example, if $f(x)=x$ for $a\leq x\leq b$ only, then the extrema of $f$ are simply $a$ and $b$.
The prior example also proved that $B$ is false for $a$ and $b$ not equal to one another.
The function $f(x)=x$ for $a\leq x\leq b$ only also satisfies C, as its derivative is always positive.
