Let f be any continuous function on [0,2]


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

  1. $\mathrm{f}^{\prime \prime}(\mathrm{x})=0$ for all $\mathrm{x} \in(0,2)$

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

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

  4. $\mathrm{f}^{\prime \prime}(\mathrm{x})>0$ for all $\mathrm{x} \in(0,2)$

Correct Option: , 2


$f(0)=0 \quad f(1)=1$ and $f(2)=2$

Let $\mathrm{h}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}$ has three roots

By Rolle's theorem $h^{\prime}(x)=f^{\prime}(x)-1$ has at least two roots

$\mathrm{h}^{\prime \prime}(\mathrm{x})=f^{\prime \prime}(\mathrm{x})=0$ has at least one roots

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