Question:
Let $\mathrm{f}$ be a twice differentiable function on $(1,6)$. If $f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1$ and $f^{\prime \prime}(x) \geq 4$, for all $x \in(1,6)$, then :
Correct Option: , 3
Solution:
$f(2)=8, f^{\prime}(2)=5, f^{\prime}(x) \geq 1, f^{\prime \prime}(x) \geq 4, \forall x \in(1,6)$
$f^{\prime \prime}(x)=\frac{f^{\prime}(5)-f^{\prime}(2)}{5-2} \geq 4 \Rightarrow f^{\prime}(5) \geq 17$ ..........(1)
$f^{\prime}(x)=\frac{f(5)-f(2)}{5-2} \geq 1 \Rightarrow f(5) \geq 11$ .............(2)
$\overline{f^{\prime}(5)+f(5) \geq 28}$
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