Answer is: option1
\( \frac{1}{16} \)Solution:
Rate of change of \( f \) at \( x = c \) is \( f'(c) \)
Rate of change of \( f \) at \( x = 1 \) is \( f'(1) \)
\( f'(c) = 4 f'(1) \)
\( f'(x) = \frac{1}{2\sqrt{2x}} \cdot 2 = \frac{1}{\sqrt{2x}} \)
\( \frac{1}{\sqrt{2c}} = 4 \cdot \frac{1}{\sqrt{2}} \)
\( \frac{1}{2c} = 16 \cdot \frac{1}{2} = 8 \)
\( c = \frac{1}{16} \) Ans