This is third part of multipart q2

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- Let $f:[a,b]\rightarrow R$ be continuous on [a,b] and differentiable on (a,b).Such that $f(a)=f(b)$ where a and b are some real numbers.Then there exists some $c$ in $(a,b)$ such that $f'(c)=0$

Step 1:

$f(x)=(x^2-1)$

$f(1)=1-1=0$

$f(2)=2^2-1=3$

$f(a)\neq f(b)$

Though it is continuous and derivable in the interval $[1,-2]$ and Rolle's theorem is not applicable.

Step 2:

$f(x)=x^2-1$

$f'(x)=2x$

$f'(x)$ is not zero in the interval $[1,2]$.

So $f(2)\neq f'(2)$

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