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Let f be a function such that f(x+y)=f(x...

Let `f` be a function such that `f(x+y)=f(x)+f(y)` for all `x` and `y` and `f(x)=(2x^2+3x)g(x)` for all `x `, where `g(x)` is continuous and `g(0)=3`, then find `f^(prime)(x)`.

Text Solution

Verified by Experts

The correct Answer is:
9
`f'(x)=underset(hrarr0)lim(f(x+h)-f(x))/(h)`
`=underset(hrarr0)lim(f(x)+f(h)-f(x))/(h)`
`=underset(hrarr0)lim(f(h))/(h)`
`=underset(hrarr0)lim((2h^(2)+3h)g(h))/(h)`
`=underset(hrarr0)lim(2h+3)g(h)`
`=(0+3)g(0)`
`=3g(0)`
`=3xx3`
`=9`
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