Question

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

• A. $f^{\prime }\left(1\right)>1$
• B. $f^{\prime }\left(1\right)\le 0$
• 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 A $f^{\prime }\left(1\right)>1$

$f^{\prime \prime }\left(x\right)>0$, $f\left(\frac{1}{2}\right)=\frac{1}{2}$ and $f\left(1\right)=1$
$f^{\prime }\left(x\right)$ is always increasing

$f^{\prime }\left(1\right)>\frac{f\left(1\right)-f\left(\frac{1}{2}\right)}{1-\frac{1}{2}}$
$f^{\prime }\left(1\right)>1$
Slope of tangent at $B>$ Slope of chord $AB$.