Which of the following functions is not differentiable at x = 0? f(x) = x^2 sin(
Practice Questions
Q1
Which of the following functions is not differentiable at x = 0? f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0.
f(x)
g(x) =
x
Questions & Step-by-Step Solutions
Which of the following functions is not differentiable at x = 0? f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0.
Step 1: Understand what it means for a function to be differentiable at a point. A function is differentiable at a point if it has a defined derivative at that point.
Step 2: Look at the function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0. We need to check if the derivative exists at x = 0.
Step 3: Calculate the derivative of f(x) as x approaches 0. The derivative is defined as the limit of (f(x) - f(0)) / (x - 0) as x approaches 0.
Step 4: Substitute f(0) = 0 into the limit: we need to evaluate the limit of (x^2 sin(1/x)) / x as x approaches 0.
Step 5: Simplify the limit to (x sin(1/x)). As x approaches 0, sin(1/x) oscillates between -1 and 1, causing the limit to not settle on a single value.
Step 6: Since the limit does not exist, the derivative at x = 0 does not exist, meaning f(x) is not differentiable at x = 0.
