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

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Continuity and Differentiability

The function

f (x) = x^{2} - 2 x - 3 \cdot e^{∣ 9 x^{2} - 12 x + 4 ∣}

is not differentiable at exactly:

Answer: (C)

The factor

x^{2} - 2 x - 3 = (x - 3) (x + 1)

has simple zeros at

x = - 1

and

x = 3

, where

∣ x^{2} - 2 x - 3∣

has corners. The exponent

9 x^{2} - 12 x + 4 = (3 x - 2)^{2}

is a perfect square, so

∣9 x^{2} - 12 x + 4∣ = (3 x - 2)^{2}

is smooth and

e^{(3 x - 2)^{2}}

is differentiable everywhere and never zero. The product therefore fails to be differentiable only at

x = - 1

and

x = 3

, that is, at two points.

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option