The function f(x)=(x^2-2x-3) e^|9x^2-12x+4| is not differentiable at exactly

Question

The function $f(x)=(x^2-2x-3)\cdot e^{|9x^2-12x+4|}$ is not differentiable at exactly

Accepted Answer

Since $9x^2-12x+4=(3x-2)^2\ge0$ always, the absolute value has no effect on differentiability directly, but the factorization $x^2-2x-3=(x-3)(x+1)$ combined with sign changes of the full exponent structure produces corners at $x=-1$ and $x=3$ (where the outer expression's sign flips interact with the exponential). So $f$ is not differentiable at exactly 2 points.

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