Question

The number of points where the function \(f(x)=\left(x^{2}-1\right)\left|x^{2}-x-2\right|+\sin (|x|)\) is not differentiable is (A) 0 (B) \(I\) (C) 2 (D) 3

Short answer

Answer: (C) 2

Step-by-step solution

Identify critical points for each term separately

First, we need to identify the critical points for each term, i.e., the points where the function might be non-differentiable. We have three terms in the function: a quadratic term, an absolute value term, and a sine term. In the function \(f(x)\), - Quadratic term: \(x^2 - 1\) is differentiable everywhere - Sine term: \(\sin(|x|)\) is differentiable everywhere For the absolute value term, let's break it down further: - Absolute value term: \(\left|x^2 - x - 2\right|\) is not differentiable at the points where \(x^2 - x - 2 = 0\)

Find the roots of the polynomial x^2 - x - 2

To find the points where the absolute value term might be non-differentiable, we need to find the roots of the polynomial \(x^2 - x - 2\). We can do this by using the quadratic formula: \(x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a=1, b=-1\) and \(c=-2\) \(x =\frac{1 \pm \sqrt{1+8}}{2} = \frac{1 \pm \sqrt{9}}{2} = \frac{1 \pm 3}{2}\) The roots are \(x=2\) and \(x=-1\)

Check if the function is non-differentiable at the critical points

Now, we have two critical points: \(x=2\) and \(x=-1\). We need to check if the function is indeed non-differentiable at these points. We form the complete function: \(f(x)=\left(x^{2}-1\right)\left|x^{2}-x-2\right|+\sin (|x|)\) We observe that at these points, the function is continuous, but its derivative changes as it crosses these points due to the presence of \(\left|x^2-x-2\right|\). Hence, at \(x=2\) and \(x=-1\), the function has non-differentiable points. Therefore, there are 2 non-differentiable points in the function \(f(x)\)

Compare the result with the provided options

Our result is that there are 2 non-differentiable points in the function \(f(x)=\left(x^{2}-1\right)\left|x^{2}-x-2\right|+\sin (|x|)\). Comparing it with the given options, we find out that the correct answer is: (C) 2