First slide

Differentiability

Question

$If f (x) f (y) = f (x + y) for all x, y, suppose f (5) = 2 and f^{1} (0) = 3,$ $t h e n f (5) i s e q u a l t o$

Moderate

A

0
B

5
C

6
D

11

Solution

$\begin{array}{l} f (x + y) = f (x) f (y) \\ \Rightarrow f (0 + 5) = f (0) f (5) \Rightarrow f (0) = 1 [∴ f (5) = 2 \neq 0] \\ A l s o, f (5) = \lim_{h \to 0} \frac{f (5 + h) - f (5 + 0)}{h} \\ = \lim_{h \to 0} \frac{f (5) f (h) - f (5) f (0)}{h} \\ = f (5) \lim_{h \to 0} \frac{f (h) - f (0)}{h} \\ = f (5) f^{1} (0) = 2 \times 3 = 6 \\ H e n c e (1) i s t h e c o r r e c t a n s w e r . \end{array}$

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