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## Homework Statement

Hey, I'm revising my calculus 1 test, I'm just not sure if this is a valid way to prove this.

If f is continuous on [a,b] and if f'(x)=0 for every x in (a,b), prove that f(x)=k for some real number k

## Homework Equations

Rolle's theorem: if f(a)=f(b), then there is some number c such that f'(c)=0

Possibly other theorems like MVT, IVT whatever

## The Attempt at a Solution

Since f is continuous on [a,b] and differentiable on (a,b), Rolle's theorem applies.

If f'(x)=0 for every x in (a,b), then there is a number Z in the interval [c,d], where a[tex]\leq[/tex]c[tex]\leq[/tex]d[tex]\leq[/tex]b, such that (f(d)-f(c))/(d-c)=f'(Z)

Since Z is in [a,b] f'(Z)=0, so for any interval [c,d] (f(d)-f(c))/(d-c)=0, therefore f(c)=f(d) which means f(x)=k for some real number k

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