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"Let "g: R rarrR be a differentiable fu...

`"Let "g: R rarrR ` be a differentiable function satisfying `g(x)=g(y) g(x-y)AA x, y in R and g'(0)= a and g'(3)=b," Then find the value of "g'(-3).`

Text Solution

Verified by Experts

The correct Answer is:
`(a^(2))/(b)`
`g(x)=g(y)g(x-y)`
Differentiating w.r.t. x, keeping y constant,
`g'(x)=g(y)[g'(x-y)]`
Put y=x. Then,
`g'(x)=g(x)cdotg'(0)=acdotg(x)`
`"or "g(x)=ae^(x)" "[becauseg(0)=1]`
`"or "g'(x)=ae^(x),g'(3)=ae^(3)=b`
`"or "g'(-3)=ae^(-3)=(a^(2))/(b)`
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