Q125 / 200 · Manipal (MET) 2014 · Mathematics

If $f(x+y)=f(x) \cdot f(y)$ for all $x$ and $y$ and $f(15)=2, f^{\prime}(0)=3$, then $f^{\prime}(5)$ will be

  • A
    2
  • B
    4
  • Correct answer
    C
    6
  • D
    8

Solution

Let $x=5, y=0 \Rightarrow f(5+0)=f(5) \cdot f(0)$ $\begin{aligned} \Rightarrow \quad f(5) & =f(5) f(0) \Rightarrow f(0)=1 \\ \therefore \quad f^{\prime}(5) \lim _{h \rightarrow 0} & =\frac{f(5+h)-f(5)}{h} \\ & =\lim _{h \rightarrow 0} \frac{f(5) f(h)-f(5)}{h} \\ & =\lim _{h \rightarrow 0} 2\left(\frac{f(h)-1}{h}\right) \quad[\therefore f(5)=2] \\ & =2 \lim _{h \rightarrow 0}\left(\frac{f(h)-f(0)}{h}\right)=2 \times f(0) \\ & =2 \times 3=6 \end{aligned}$

Hence, the correct answer is option: 'C'.

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