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KVPY2014MathematicsDefinite Integration

A continuous function f: R R satisfies the equation f(x)=x+ ₀^ x f(t) d t . Which of the following options is true ?

Options

  1. Af(x+y)=f(x)+f(y)
  2. Bf(x+y)=f(x) f(y)
  3. Cf(x+y)=f(x)+f(y)+f(x) f(y)
  4. Df(x+y)=f(x y)

Correct answer

C. f(x+y)=f(x)+f(y)+f(x) f(y)

Step-by-step solution

f^ (x)=1+f(x) f(0)=0 f(x) 1+f(x) =1 (1+f(x))=x then f(x+y)=f(x)+f(y) f(x) f(y)

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