Question

The function $f\left(x\right)=\{\begin{array}{cc}2x^2-1,& \text{if}1\le x\le 4\\ 151-30x,& \text{if}4<x\le 5\end{array}$ is not suitable to apply Rolle's theorem, since

• A. $f\left(x\right)$ is not continuous on $[1,5]$
• B. $f\left(1\right)\ne f\left(5\right)$
• C. $f\left(x\right)$ is continuous only at $x=4$
• D. $f\left(x\right)$ is not differentiable in $(4,5)$
• E. $f\left(x\right)$ is not differentiable at $x=4$

Answer 1

The correct option is E $f\left(x\right)$ is not differentiable at $x=4$

$f\left(x\right)=2x^2-1,$ if $1\le x\le 4$

$f\left(x\right)=151-30x,$ if $4<x\le 5$

For applying Rolle's Theorem, function must be differentiable in the interval.

For interval $1\le x\le 4$

$f^{\prime }\left(x\right)=4x$

or, $f^{\prime }\left(4^{-}\right)=16$

For interval $4<x\le 5$

$f^{\prime }\left(x\right)=-30$

or, $f^{\prime }\left(4^{+}\right)=-30$

here, $f^{\prime }\left(4^{-}\right)\ne f^{\prime }\left(4^{+}\right)$

$f\left(x\right)$ is not differentiable at $x=4.$

Hence, E is the correct option.