Question

If $f$ be continuous on $[1,5]$ and differentiable $(1,5)$. If $f\left(1\right)=-3$ and $f\text{'}\left(x\right)\ge 9,\forall x\in (1,5)$, then

• A. $f\left(5\right)\ge 33$
• B. $f\left(5\right)\ge 36$
• C. $f\left(5\right)\le 36$
• D. $f\left(5\right)\ge 9$
• E. $f\left(5\right)\le 9$

Answer 1

The correct option is A

$f\left(5\right)\ge 33$

Explanation for the correct option:

Given, $f\left(1\right)=-3$ and $f\text{'}\left(x\right)\ge 9,\forall x\in (1,5)$

By applying Lagrange mean value theorem, we get

$f\left(x\right)=\frac{f\left(5\right)-f\left(1\right)}{5-1}\ge 9$

$\Rightarrow$ $\frac{f\left(5\right)+3}{4}\ge 9$

$\Rightarrow$ $f\left(5\right)\ge 36-3$

$\mathbf{\therefore }\mathit{f}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{}\mathbf{\ge }\mathbf{}\mathbf{33}$

Hence, Option ‘A’ is Correct.