Question

Let f(x+y) = f(x).f(y) for all x and y. Given that f(3) = 3 and f'(0)= 11. Then the value of f'(3) is ___ .

Answer 1

$f(x+y)=f\left(x\right).f\left(y\right)f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(x+h)-f\left(x\right)}{h}={lim}_{h\to 0}\frac{f\left(x\right).f\left(h\right)-f\left(x\right)}{h}=f\left(x\right).{lim}_{h\to 0}\frac{f\left(h\right)-1}{h}......\left(i\right)$
Since $f^{\prime }\left(x\right)$ is finite at least one point,
${lim}_{h\to 0}\frac{f\left(h\right)-1}{h}$ is finite
$\Rightarrow {lim}_{h\to 0}f\left(h\right)-1=0\Rightarrow {lim}_{h\to 0}f\left(h\right)=1$
Since f(x) is differentiable at x=0, it is also continuous
${lim}_{h\to 0}f\left(h\right)=f\left(0\right)=1f^{\prime }\left(0\right)={lim}_{h\to 0}\frac{f(0+h)-f\left(0\right)}{h}={lim}_{h\to 0}\frac{f\left(h\right)-1}{h}{f}^{\prime }\left(x\right)=f\left(x\right){lim}_{h\to 0}\frac{f\left(h\right)-1}{h}.....from\left(i\right)f^{\prime }\left(x\right)=f\left(x\right).f^{\prime }\left(0\right)f^{\prime }\left(3\right)=f\left(3\right).f^{\prime }\left(0\right)=11f\left(3\right)=11\times 3=33[\because f^{\prime }(0)=11]$