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  4. If f is differentiable, f(x+y)=f(x) f(y) for all x, y ∈ I R, f(3)=3, f prime(0)=11, then f prime(3)=

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Q. If $f$ is differentiable, $f(x+y)=f(x) f(y)$ for all $x, y \in I R, f(3)=3, f^{\prime}(0)=11$, then $f^{\prime}(3)=$

TS EAMCET 2017

Solution:

We have,
$f(x+y)=f(x) f(y)$
differentiate with respect to $x, y$ as constant, we get
$f'(x+y)=f'(x) f(y)$
Put $x=0, y=3$, we get
$ f'(0+3) =f'(0) f(3) $
$ \Rightarrow f'(3) =f'(0) \cdot f(3)$
$\Rightarrow f'(3)=11 \times 3=33$
$[\because f'(0)=11, f(3)=3]$