The function $f (x) = | x^{2} - 3 x + 2 | + c o s | x |$ is not differentiable at x is equal to -

- 1

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Solution

The correct option is D 2
$∵ f (x) = | x^{2} - 3 x + 2 | + c o s | x | = | (x - 1) | | x - 2 | + c o s | x | f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x^{2} - 3 x + 2 + c o s x, & x < 0 x^{2} - 3 x + 2 + c o s x, & 0 \leq x < 1 - x^{2} + 3 x - 2 + c o s x, & 1 \leq x < 2 x^{2} - 3 x + 2 + c o s x, & x > 2 \end{matrix} ∴ f^{^{'}} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 x - 3 - s i n x, & x < 0 2 x - 3 - s i n x, & 0 \leq x < 1 - 2 x + 3 - s i n x, & 1 \leq x < 2 2 x - 3 - s i n x, & x > 2 \end{matrix}$
It is clear f(x) is not differentiable at x = 1and x = 2.
$∴ f^{^{'}} (1^{-}) = - 1 - s i n 1 a n d f^{^{'}} (1^{+}) = 1 - s i n 1$
$∴ f^{^{'}} (2^{-}) = - 1 - s i n 2 a n d f^{^{'}} (2^{+}) = 1 - s i n 2$

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