Question:
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
Correct Option: , 2
Solution:
$f(0)=0 \quad f(1)=1$ 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