Question

Let \(f(x)=\max\left\{x^{2},x^{3}\right\}\) for all \(x\in\mathbf{R}\). Then which of the following is NOT true ?

• A. \(f\) is continuous for all \(x\).
• B. \(f\) is not differentiable exactly at one point in \(\mathbf{R}\).
• C. \(f\) is not differentiable exactly at two points in \(\mathbf{R}\).
• D. \(f'\left(-\frac{1}{2}\right)+f'\left(\frac{1}{8}\right)+f'\left(\frac{3}{2}\right)=6\)

Hint:

Write \(f\) piecewise: \(x^2\) for \(x\le 1\) and \(x^3\) for \(x\ge 1\), then test each statement.

Answer 1

The Correct Option is C

Compare \(x^2\) and \(x^3\): \(x^3-x^2=x^2(x-1)\), which is \(\le 0\) for \(x\le 1\) and \(\ge 0\) for \(x\ge 1\).
So \(f(x)=x^2\) for \(x\le 1\) and \(f(x)=x^3\) for \(x\ge 1\).
(1) Both pieces are continuous and equal to 1 at \(x=1\); so \(f\) is continuous everywhere: true.
(2) At \(x=1\): left derivative \(=2\), right derivative \(=3\), not equal, so \(f\) is not differentiable at \(x=1\). At \(x=0\), \(f(x)=x^2\) on both sides, so \(f\) is differentiable. Hence \(f\) fails to be differentiable at exactly one point: true.
(3) Not differentiable at exactly two points: false (only \(x=1\)).
(4) \(f'(-\tfrac12)=2(-\tfrac12)=-1\), \(f'(\tfrac18)=2\cdot\tfrac18=\tfrac14\), \(f'(\tfrac32)=3\cdot\tfrac94=\tfrac{27}{4}\); the sum is \(-1+\tfrac14+\tfrac{27}{4}=6\): true.
The statement that is NOT true is (3).
Answer: (3) \(f\) is not differentiable exactly at two points in \(\mathbf{R}\).