The number of points, where the function f:R→R,f(x),=|x−1|cos⁡|x−2|sin ⁡|x−1|+(x−3)x2−5x+4, is NOT differentiable, is :

f(x)=|x−1|cos⁡|x−2|sin⁡|x−1|+(x−3)x2−5x+4=|x−1|cos⁡|x−2|sin⁡|x−1|+(x−3)|x−1||x−4|=|x−1|[cos⁡|x−2|sin⁡|x−1|+(x−3)|x−4|]which is not differentiable at x=1 and x=4The number of points at which f(x) is not differentiable =2

The number of points, where the function f:R→R,f(x),=|x−1|cos⁡|x−2|sin ⁡|x−1|+(x−3)x2−5x+4, is NOT differentiable, is :

f(x)=|x−1|cos⁡|x−2|sin⁡|x−1|+(x−3)x2−5x+4=|x−1|cos⁡|x−2|sin⁡|x−1|+(x−3)|x−1||x−4|=|x−1|[cos⁡|x−2|sin⁡|x−1|+(x−3)|x−4|]which is not differentiable at x=1 and x=4The number of points at which f(x) is not differentiable =2

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The number of points, where the function $f : R \to R, f (x)$ , $= | x - 1 | \cos | x - 2 | \sin | x - 1 | + (x - 3) |x^{2} - 5 x + 4|$ , is NOT differentiable, is :

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answer is B.

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Detailed Solution

$f (x) = | x - 1 | \cos | x - 2 | \sin | x - 1 | + (x - 3) |x^{2} - 5 x + 4|$
$\begin{array}{l} = | x - 1 | \cos | x - 2 | \sin | x - 1 | + (x - 3) | x - 1 | | x - 4 | \\ = | x - 1 | [\cos | x - 2 | \sin | x - 1 | + (x - 3) | x - 4 |] \end{array}$
which is not differentiable at $x = 1 and x = 4$