Question

If $f(x + y) = f(x).f(y)$ for all x and y and $f^{'}(0) = 3,$ then $f^{'}(5)$ will be

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (C)

6

Answer 2

Let $x = 5,y = 0$ ⇒ $f(5 + 0) = f(5).f(0)$

⇒ $f(5) = f(5)f(0)$ ⇒ $f(0) = 1$

Therefore, $f^{'}(5) = \lim_{h \rightarrow 0}\frac{f(5 + h) - f(5)}{h}$

= $\lim_{h \rightarrow 0}\frac{f(5)f(h) - f(5)}{h} = \lim_{h \rightarrow 0}2\left\lbrack \frac{f(h) - 1}{h} \right\rbrack\{\because f(5) = 2\}$

= $2\lim_{h \rightarrow 0}\left\lbrack \frac{f(h) - f(0)}{h} \right\rbrack = 2 \times f^{'}(0) = 2 \times 3 = 6$.