Let $f: R→R$ be defined as $f(x) = |x|+|x^2-1|$. The total number of points at which f attains either a local maximum or a local maximum, is
Answer & explanation
Correct answer: option 1
The correct answer is option (1) : 5
We have,
$f(x) = |x|+|x^2-1|=\left\{\begin{matrix}-x+x^2-1,&x ≤-1\\-x-x^2+1,&-1x< 0\\x-x^2 +1, & 0≤x < 1\\x+x^2-1, & x ≤ 1 \end{matrix}\right.$
$∴f'(X) = \left\{\begin{matrix}2x-1, &x<-1\\-2x-1,&-1< x, 0\\-2x+1,&0< x< 1\\2x+1,& x> 1\end{matrix}\right.$
Clearly, f(x) is not differentiable at x = -1, 0, 1.
The changes in signs of f' (x) for different values of x are shown in figure.
So, f'(x) changes its sign at 5 points.
Hence, total number of points of local maximum or local minimum of f(x) is 5.