Mathematics - Continuity - Differentiability Question with Solution | TestHub

Given that$f\left(x\right)=\left|x^2-2x-3\right|e^{\left(9x^2-12x+4\right)}$

Differentiating on both sides

$f^{\text{'}}\left(x\right)=\left|x^2-2x-3\right|.e^{\left(9x^2-12x+4\right)}.\left(18x-12\right) + \frac{\left|x^2-2x-3\right|}{x^2-2x-3}.\left(2x-2\right).e^{\left(9x^2-12x+4\right)}$

$\Rightarrow f\text{'}\left(x\right) = \left|x^2-2x-3\right|.e^{\left(9x^2-12x+4\right)} \left\{\left(18x-12\right)+\frac{\left(2x-2\right)}{x^2-2x-3}\right\}$

$\Rightarrow f\text{'}\left(x\right) = \left|x^2-2x-3\right|.e^{\left(9x^2-12x+4\right)}.\frac{\left\{18x^3-36x^2-54x-12x^2+24x+36+2x-2\right\}}{x^2-2x-3}$

$\Rightarrow f\text{'}\left(x\right) = \left|x^2-2x-3\right|.e^{\left(9x^2-12x+4\right)}.\frac{\left\{18x^3-48x^2-28x+34\right\}}{\left(x-3\right)\left(x+1\right)}$

Here $f\text{'}\left(x\right)$ is not defined at $x = -1 \& x=3$.

$f\left(x\right)$is not differentiable at two points.