Question

Let \(f:\mathbb{R}\to\mathbb{R}\) be a function defined by \(f(x)=\max\{x,x^2\}\). Let S denote the set of all points in \(\mathbb{R}\), where f is not differentiable. Then :

• A. \(S=\{0,1\}\)
• B. \(S=\{0\}\)
• C. \(S=\phi\) (an empty set)
• D. \(S=\{1\}\)

Hint:

Find where \(x=x^2\) and check one-sided derivatives there.

Answer 1

The Correct Option is A

\(x^2\ge x\) when \(x\le0\) or \(x\ge1\), and \(x\ge x^2\) when \(0\le x\le1\).
So \(f(x)=x^2\) for \(x<0\), \(f(x)=x\) for \(0\le x\le1\), \(f(x)=x^2\) for \(x>1\).
At \(x=0\): left derivative \(=2x=0\), right derivative \(=1\) (not equal).
At \(x=1\): left derivative \(=1\), right derivative \(=2x=2\) (not equal).
So f is not differentiable at exactly 0 and 1, \(S=\{0,1\}\).
Answer: (1) \(S=\{0,1\}\)