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Let $f(x)=4x^3-3x^2-2x+1,$ use Rolle's theorem to prove that there exist $c,0<c<1$ such that $f(c)=0$


$f(x)=4x^3-3x^2-2x+1,$ is continuous and differentiable but the third condition of Rolle's theorem is not met,so Rolle's theorem is not applicable here.

I am stuck here.

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  • $\begingroup$ $f(c)=0$ or $f'(c)=0$ $\endgroup$ – user223391 Jun 22 '17 at 17:06
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Let $g(x)=x^4-x^3-x^2+x$. Then $g'(x)=f(x)$.

$g(0)=g(1)=0$.

By Rolle's Theorem, there exist $c$ such that $0<c<1$ and $g'(c)=0$.

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