Question

Let \( f(x) = x^2 \log(\cos x) \log(1 + x) \) for \( x \neq 0 \), and \( f(0) = 0 \). Determine the behavior of \( f(x) \) at \( x = 0 \).

• A. not continuous
• B. continuous but not differentiable
• C. differentiable
• D. not continuous, but differentiable

Hint:

To check if a function is differentiable at a point, verify both its continuity and the existence of the derivative at that point.

Answer 1

The Correct Option is C

Step 1: {Check for continuity at \( x = 0 \)}
We need to find the limit of \( f(x) \) as \( x \to 0 \): \[ \lim_{x \to 0} \frac{x^2 \log(\cos x)}{\log(1 + x)} = \lim_{x \to 0} x^2 \cdot \log(\cos x) = 0 \cdot \log(1) = 0 \] Thus, \( f(x) \) is continuous at \( x = 0 \). 
Step 2: {Check for differentiability at \( x = 0 \)}
We now check if \( f(x) \) is differentiable at \( x = 0 \) by finding the derivative at this point: \[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{h^2 \log(\cos h)}{h \log(1 + h)} = 0 \] Thus, \( f(x) \) is differentiable at \( x = 0 \).