Editing Rolle's theorem (section)

===Functions with zero derivative===

Rolle's theorem implies that a [[differentiable function]] whose derivative is {{tmath|0}} in an interval is constant in this interval. 

Indeed, if {{mvar|a}} and {{mvar|b}} are two points in an interval where a function {{mvar|f}} is differentiable, then the function
<math display=block>g(x)=f(x)-f(a)-\frac{f(b)-f(a)}{b-a}(x-a)</math>
satisfies the hypotheses of Rolle's theorem on the interval {{tmath|[a,b]}}.

If the derivative of {{tmath|f}} is zero everywhere, the derivative of {{tmath|g}} is
<math display=block>g'(x)=\frac{f(b)-f(a)}{b-a},</math>
and Rolle's theorem implies that there is {{tmath|c\in (a,b)}} such that
<math display=block>0=g'(c)=\frac{f(b)-f(a)}{b-a}.</math>

Hence, {{tmath|1=f(a)=f(b)}} for every {{tmath|a}} and {{tmath|b}}, and the function {{tmath|f}} is constant.