Differentiability

Using the trigonometric identity: we rewrite the integral as: Simplify: 1. Evaluate : At the limits: 2. Evaluate : At the limits: Substitute back: Final Answer:

The function consists of absolute value functions. - For , , which is continuous and differentiable. - For , , which is continuous but not differentiable at . Thus, is continuous but not differentiable at both and .

To check the differentiability of the function at , we need to check if the derivative exists at that point. 1. The function is defined as 2. The derivative of for is For , the derivative is 3. At , we need to check if the left-hand derivative and the right-hand derivative exist and are equal: - Left-hand derivative: - Right-hand derivative: Since the left-hand and right-hand derivatives are not equal, the derivative does not exist at . Therefore, the function is not differentiable at .