Mathematics - Continuity - Differentiability Question with Solution | TestHub

The function $f(x)$ is a product of differentiable functions $x^2$, $e^{ax}$ and $|x-c|$. The only point where differentiability might fail is $x=c$ due to $|x-c|$. For $f(x)$ to be differentiable at $x=c$, the left-hand derivative (LHD) and right-hand derivative (RHD) must be equal. We have $f(x)=x^2(c-x)e^{ax}$ for $x<c$ and $f(x)=x^2(x-c)e^{ax}$ for $x\ge c$. Calculating the derivatives, we get LHD at $x=c$ as $-c^2{e}^{ac}$ and RHD at $x=c$ as $c^2{e}^{ac}$. For differentiability, $-c^2{e}^{ac}=c^2{e}^{ac}$, which implies $2c^2{e}^{ac}=0$. Since $e^{ac}\ne 0$, we must have $c^2=0$, so $c=0$.