First slide

Differentiability

Question

$Let f^{'} (x) be continuous at x = 0 and f^{''} (0) = 4 . The value of$ $\lim_{x \to 0} \frac{2 f (x) - 3 f (2 x) + f (4 x)}{x^{2}}$

Moderate

A

2
B

11
C

12
D

None of these

Solution

$\begin{array}{l} \lim_{x \to 0} \frac{2 f (x) - 3 f (2 x) + f (4 x)}{x^{2}} [F o r m \frac{0}{0}] \\ = \lim_{x \to 0} \frac{2 f' (x) - 6 f' (2 x) + 4 f' (4 x)}{2 x} [F o r m \frac{0}{0}] \\ = \lim_{x \to 0} 2 f'' (x) - 12 f " (2 x) + 16 f " (4 x) \\ = f " (0) - 6 f " (0) + g f " (0) = 3 f " (0) = 3 \times 4 = 12 \end{array}$
$Hence (3)is the correct answer.$

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