Suppose That f is A Differentiable Function With The Property That F(x + y) = f(x) + f(y) + xy and

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Question

Suppose that f is a differentiable function with the property that

f(x + y) = f(x) + f(y) + xy and

easy

Solution

Correct option is

f(x) = 3x + x²/2

Hence f(x) = 3x + x²/2 + C. Putting x = y = 0 in the given equation, we have f(0) = f(0 + 0) = f(0) + f(0) + 0 ⇒ f(0) = 0. Thus C = 0 and f(x) = 3x+ x²/2.