Question

Let ƒ be a function such that ƒ(x + y) = ƒ(x) + ƒ(y) for all x and y and ƒ(x) = (2x² + 3x) g(x) for all x where g(x) is continuous and g(0) = 3. Then ƒ′(x) is equal to –

• A. 9
• B. 3
• C. 6
• D. None of these

Answer 1

Answer: (A)

9

Answer 2

ƒ′(x) = $\operatorname { Lim } _ { \mathrm { h } \rightarrow 0 }$ $\frac{ƒ(x + h)–ƒ(x)}{h}$

= $\operatorname { Lim } _ { \mathrm { h } \rightarrow 0 }$ $\frac{ƒ(x) + ƒ(h)–ƒ(x)}{h}$

= $\operatorname { Lim } _ { \mathrm { h } \rightarrow 0 }$ $\frac{ƒ(h)}{h}$

= $\operatorname { Lim } _ { \mathrm { h } \rightarrow 0 }$ $\frac{(2h^{2} + 3h)g(h)}{h}$ = $\underset{h \rightarrow 0}{Lim}$ (2h + 3) g(h)

= (0 + 3) g(0)

= 3g (0)

= 3.3

= 9.