General — Hard JEE math MCQ
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a twice differentiable function such that $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}$, and $f\left(\frac{1}{2}\right)=\frac{1}{2}, f(1)=1$, then
- A. $f^{\prime}(1) \leq 0$
- B. $0<f^{\prime}(1) \leq \frac{1}{2}$
- C. $\frac{1}{2}<f^{\prime}(1) \leq 1$
- D. $f^{\prime}(1)>1$
Solution
The correct option is **D**.