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Question

Suppose a differentiable function f(x) satisfies the identity f(x+y)=f(x)+f(y)+xy2+x2y for all real x and y. If limx0f(x)x=1, then f(3) is equal to

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Solution

Given: f(x+y)=f(x)+f(y)+xy2+x2y
At x=y=0
f(0)=2f(0)f(0)=0
Now,
f(x)=limh0f(x+h)f(x)h
(take y=h)
f(x)=limh0f(h)h+limh0(xh)+x2
f(x)=1+0+x2
f(x)=1+x2
f(3)=10

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