Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?

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Question: Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?

Options:

Correct Answer: Yes

Solution:

Using the limit definition, f\'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0)/h] = 0. Thus, f(x) is differentiable at x = 0.

Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?

Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?

Correct Answer: Yes, f(x) is differentiable at x = 0.

Step 1: Understand the function f(x) = x^2 sin(1/x). This function is defined for all x except x = 0, but we need to check its behavior at x = 0.
Step 2: To check if f(x) is differentiable at x = 0, we use the limit definition of the derivative: f'(0) = lim (h -> 0) [(f(h) - f(0)) / h].
Step 3: Calculate f(0). Since f(x) is not defined at x = 0, we define f(0) = 0 for the purpose of this limit.
Step 4: Substitute f(h) into the limit: f(h) = h^2 sin(1/h). So, we have f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0) / h].
Step 5: Simplify the expression: f'(0) = lim (h -> 0) [h sin(1/h)].
Step 6: Analyze the limit. As h approaches 0, sin(1/h) oscillates between -1 and 1, so h sin(1/h) will approach 0 because h approaches 0.
Step 7: Conclude that f'(0) = 0. Since the limit exists and is finite, f(x) is differentiable at x = 0.

Differentiability – The question tests the understanding of whether a function is differentiable at a specific point using the limit definition of the derivative.
Limit Definition of Derivative – The solution requires applying the limit definition of the derivative to evaluate the differentiability of the function at x = 0.
Behavior of Functions at Singular Points – The function involves a term sin(1/x), which can lead to confusion regarding its behavior as x approaches 0.