Application of Derivatives 3 Question 1
####1. If $f: R \rightarrow R$ is a twice differentiable function such that $f^{\prime \prime}(x)>0$ for all $x \in R$, and $f \frac{1}{2}=\frac{1}{2}, f(1)=1$, then
(a) $f^{\prime}(1) \leq 0$
(b) $f^{\prime}(1)>1$
(c) $0<f^{\prime}(1) \leq \frac{1}{2}$
(d) $\frac{1}{2}<f^{\prime}(1) \leq 1$
(2017 Adv.)
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Answer:
Correct Answer: 1. (b)
Solution:
- $f^{\prime}(x)$ is increasing
For some $x$ in $\frac{1}{2}, 1$
$ f^{\prime}(x)=1 $
$ \therefore \quad f^{\prime}(1)>1 $
[LMVT]
