Let 𝑓: ℝ → ℝ be a function such that 𝑓(𝑥) = max{𝑥, 𝑥3}, 𝑥 ∈ ℝ, where ℝ is the set of all real numbers. The set of all points where 𝑓(𝑥) is NOT differentiable is

A.

{−1, 1, 2}

B.

{−2, −1, 1}

C.

{0, 1}

D.

{−1, 0, 1}

Solution:

To find the points where the function  𝑓(𝑥) = max{𝑥, 𝑥3} is not differentiable, we need to examine the function at critical points, where the maximum changes.

First, we find critical points by setting the derivative of the function equal to zero:
f'(x)=ddxmax(x, x3)

  • For x such that x>0, the function x3, and its derivative is 3x2
  • For x such that x>0, the function x, and its derivative is 1.
  • At x=0, the derivatives of both x and x3 are equal to 0.

so we can take any derivative (e.g., x) at this point. Hence, the derivative 𝑓′(x) exists for all x in ℝ.

Now, we examine the points where the maximum changes. These are the points where x=x3.

Solving x=x3 gives us solutions, x = −1,0,1.

So, the points where the function is not differentiable are x=−1,0,1.