Let mathrm{f} be any continuous function on [ | vedclass

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Question

$f(0)=0 \quad f(1)=1$ and $f(2)=2$

Let $\mathrm{h}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}$ has three roots

By Rolle's theorem $\mathrm{h}^{\pr

Vedclass · Accepted Answer

$f^{\prime \prime}(x)=0$ for some $x \in(0,2)$

Vedclass · Answer

$f(0)=0 \quad f(1)=1$ and $f(2)=2$

Let $\mathrm{h}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}$ has three roots

By Rolle's theorem $\mathrm{h}^{\prime}(\mathrm{x})=f^{\prime}(\mathrm{x})-1$ has at least two roots

$\mathrm{h}^{\prime \prime}(\mathrm{x})=f^{\prime \prime}(\mathrm{x})=0$ has at least one roots

Let $\mathrm{f}$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $\mathrm{f}(0)=0, \mathrm{f}(1)=1$ and $f(2)=2$, then

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