Check if the function \(f(x) = |x|\) is differentiable over the interval \([-2,2]\).
Answer
Finally, we check the derivative at \(x = 0\), which is the point where the function is not defined. Since the derivative does not exist at \(x = 0\), the function is not differentiable over the interval \([-2,2]\).
Step 1 ：First, we find the derivative of the function. The derivative of \(f(x) = |x|\) is \(f'(x) = x/|x|\) for \(x \neq 0\).
Step 2 ：Then, we check the derivative at the endpoints of the interval. At \(x = -2\), \(f'(-2) = -1\); at \(x = 2\), \(f'(2) = 1\).
Step 3 ：Finally, we check the derivative at \(x = 0\), which is the point where the function is not defined. Since the derivative does not exist at \(x = 0\), the function is not differentiable over the interval \([-2,2]\).
