Question

The number of points, where the function $f: \mathbb{R} \to \mathbb{R}$, $$ f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|, $$ is NOT differentiable, is 

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

To determine the points where the function f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4| is not differentiable, we need to examine each component of the function separately, particularly focusing on where differentiability might be lost due to absolute values or piecewise definitions.

1. Identify the points of potential non-differentiability:
   • Functions involving absolute values can have non-differentiability at the points where the absolute value inside becomes zero.
   • For |x - 1|, non-differentiability might occur at x = 1 since that's where the expression inside becomes zero.
   • Similarly, for |x - 2|, check differentiability at x = 2.
   • Finally, for the factor |x^2 - 5x + 4|, solve x^2 - 5x + 4 = 0:
     • This can be factored as (x - 1)(x - 4) = 0.
     • So, potential points are x = 1 and x = 4.
2. Examine the points:
   • At x = 1: Already identified from both |x-1| and |x^2 - 5x + 4|.
   • At x = 2: Check due to the |x - 2| term.
   • At x = 4: Check due to the |x^2 - 5x + 4| term.
3. Key Observations:
   • The function might be non-differentiable at x = 1 due to both terms: |x - 1| and |x^2 - 5x + 4|.
   • There is possible non-differentiability at x = 2 due to |x - 2|.
   • x = 4 is not a concern since it does not affect differentiability via the first term |x - 1|\cos|x - 2|\sin|x - 1|.

Thus, the function f(x) is not differentiable at two points: x = 1 and x = 2. Therefore, the correct answer is 2.