Question

Given the function:

\[ f(x) = \begin{cases} \frac{2x e^{1/2x} - 3x e^{-1/2x}}{e^{1/2x} + 4e^{-1/2x}}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \]

Determine the differentiability of \( f(x) \) at \( x = 0 \).

• A. \( f'(0^+) = -\frac{3}{4} \)
• B. \( f'(0^-) = 2 \)
• C. \( f(x) \) is not differentiable at \( x = 0 \)
• D. \( f(x) \) is differentiable at \( x = 0 \)

Hint:

For differentiability at \( x = c \), check if \( f'(c^+) = f'(c^-) \). If they are unequal, \( f(x) \) is not differentiable at \( x = c \).

Answer 1

The Correct Option is C

Step 1: Finding Left and Right Derivatives We compute: \[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h}, \quad f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h}. \] Evaluating both derivatives, we find: \[ f'(0^+) \neq f'(0^-). \] Thus, \( f(x) \) is not differentiable at \( x = 0 \).