Question

If $f:R\to R$ is a twice differentiable function such that $f"\left(x\right)>0$ for all $xϵR$, and $f\left(\frac{1}{2}\right)=\frac{1}{2}$, f(1) = 1, then,

• A. $f^{\prime }\left(1\right)\le 0$
• B. $f^{\prime }\left(1\right)>1$
• C. $0<f^{\prime }\left(1\right)\le \frac{1}{2}$
• D. $\frac{1}{2}<f^{\prime }\left(1\right)\le 1$

Answer 1

The correct option is B $f^{\prime }\left(1\right)>1$

f'(x) is increasing since $f"\left(x\right)>0$
For some x in $(\frac{1}{2},1)$
$f^{\prime }\left(x\right)=1\therefore f^{\prime }\left(1\right)>1$