To determine where the function \( f(x) = (x^2 - 1)|x^2 - 3x + 2| + \cos|x| \) is non-differentiable, we need to check the points where:
- The term \( |x^2 - 3x + 2| \) changes its form, which is typically at the roots or critical points of the expression inside the absolute value.
- The derivative of \( \cos|x| \) is discontinuous, which happens at the points where the absolute value function changes its behavior.
Let's analyze each part separately:
1. Analysis of \( |x^2 - 3x + 2| \):
Factorizing the quadratic expression:
\(x^2 - 3x + 2 = (x - 1)(x - 2)\)
The expression changes sign at \( x = 1 \) and \( x = 2 \). Hence, these are potential points of non-differentiability.
2. Analysis of \( \cos|x| \):
The derivative of \( \cos|x| \) with respect to \( x \) will involve the derivative of the inner function, which is discontinuous at \( x = 0 \).
Points of Interest:
- \( x = 1 \): Here, \( (x - 1)(x - 2) \) changes from negative to positive, indicating a possible non-differentiable point.
- \( x = 2 \): Another point where \( (x - 1)(x - 2) \) changes from positive to negative, indicating another possible point of non-differentiability.
- \( x = 0 \): The derivative of \( \cos|x| \) is discontinuous at this point, but checking the full expression will confirm its differentiability as \( |x^2 - 3x + 2| \) is continuous at \( x = 0 \).
Conclusion:
Upon examining these points in the expression \( f(x) \), we find:
- The function is continuous across \( x = 0 \) due to the behavior of \( |x^2 - 3x + 2| \).
- Discontinuity at \( x = 2 \) due to the change in \( (x - 1)(x - 2) \) influencing \( |x^2 - 3x + 2| \).
Therefore, the function \( f(x) \) is non-differentiable at \( x = 2 \).