= Solution
{c}
\b[A <maximum> or <minimum> can occur at an endpoint.] The rule that a <derivative> vanishes at an extremum requires an interior extremum and <differentiability> there. The hypotheses give no two-sided <derivative> at the endpoints, and even an existing one-sided <derivative> need not vanish there.
For example, $f(x)=x$ on $[0,1]$ has $m=0$ at $x_m=0$ and $M=1$ at $x_M=1$, but $f'(x)=1$ at every interior point; its endpoint one-sided <derivatives> are also one. Its graph meets both horizontal bounding lines without being tangent to either. Thus the claimed conclusion about two zeros of the <derivative> fails.
There is a second logical issue: the argument never establishes two distinct extremizing points. For a <constant function>, the <maximum> and <minimum> coincide and may be represented by the same point. The <constant function> does have many <stationary points>, but that needs its own argument; it does not follow from counting the same selected point twice. In <Rolle's theorem>, equal endpoint values ensure an interior extremum when the <function> is nonconstant, which is the missing mechanism here.
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