= Mean value theorem
{title2=$f'(c)=\frac{f(b)-f(a)}{b-a}$}
{wiki}
The <mean value theorem> says that an average rate of change is attained as an instantaneous rate somewhere between the endpoints. Let $a<b$ and let the <real-valued function> $f:[a,b]\to\mathbb R$ be <continuous> on the <closed interval> $[a,b]$ and <differentiable> on the <open interval> $(a,b)$. Then there exists $c\in(a,b)$ such that
$$
f'(c)=\frac{f(b)-f(a)}{b-a}.
$$
The right-hand side is the <slope> of the <secant line> joining $(a,f(a))$ to $(b,f(b))$; the left-hand side is the <slope> of the <tangent line> at $(c,f(c))$. The <derivative> need not be <continuous>, and the <function> need not be <differentiable> at the endpoints. https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem[OpenStax, Calculus Volume 1, Section 4.4: mean value theorem]{ref}
For the <polynomial> $f(x)=x^2$ on $[1,3]$, the <secant line> has <slope> $(9-1)/(3-1)=4$. Since the <derivative> is $f'(x)=2x$, the required interior point is $c=2$. The <tangent line> there is $y=4+4(x-2)=4x-4$, whereas the <secant line> is $y=1+4(x-1)=4x-3$. Their equal <slopes> make them parallel, but they are distinct <straight lines>: the theorem equates rates, not heights.
\Image[/mean-value-theorem-secant-and-tangent.png]
{title=The <secant line> and interior <tangent line> to $x^2$ have equal <slopes> but different heights}
The practical strength is that the unknown point $c$ can disappear from an estimate. Under the same hypotheses, if the <derivative> satisfies $|f'(x)|\leq M$ throughout $(a,b)$, where $M\geq0$, then
$$
|f(b)-f(a)|=|f'(c)|\,|b-a|\leq M|b-a|.
$$
Applying this to every subinterval proves <Lipschitz continuity> with <Lipschitz bound> $M$. For example, $f(x)=x^2$ on $[1,3]$ has $|f'(x)|\leq6$, so an input change of size at most $\delta$ changes its output by at most $6\delta$, as long as both inputs lie in that <interval>. A local <derivative> bound therefore controls finite changes without computing an <integral>. The sign of the <derivative> similarly gives the <derivative criterion for monotonicity>.
Taking $M=0$ shows that a <function> with zero <derivative> throughout a <real interval> is a <constant function>, provided it is <continuous> at any included endpoints. The <interval> condition is essential: on the <domain of a function> $(-2,-1)\cup(1,2)$, the <function> equal to zero on the left <interval> and one on the right <interval> has zero <derivative> everywhere. It is a <locally constant function>, yet not a <constant function> on the whole <domain of a function>. The <mean value theorem> cannot cross the gap.
A concise proof subtracts the <secant line>: set $m=(f(b)-f(a))/(b-a)$ and $h(x)=f(x)-f(a)-m(x-a)$. Then $h(a)=h(b)=0$, and <Rolle's theorem> gives an interior point with $h'(c)=f'(c)-m=0$. The geometric idea is to remove the average trend and find a horizontal <tangent line> in what remains.
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