A continuous function f:R \toR satisfies the equation f( x ) = x + \int^{x}_{0}f( t) dt. Which o f the following options is true?

Correct Answer is : f(x+y)=f(x)+f(y)+f(x) f(y)

f ( x+ y ) = f( x ) + f ( y ) + f ( x ) f ( y )

KVPY SX SB Exam 20141 Question Paper

Question : 13 of 120

Marks: +1, -0

A continuous function

f : R \to R

satisfies the equation

f( x ) = x + ∫↖{x}↙{0}f( t)\ dt.

Which o f the following options is true?

Solution:

f( x ) = x + ∫↖{x}↙{0}f( t)\ dt

\f′(x) = 1 +\f(x) ⇒ \f′(x)-\f(x) = 1

⇒

e^{-x}\f′(x) - \f(x) e^{-x} = e^{-x}

⇒

d/{dx}(\f(x) e^{-x}) = e^{-x}

⇒

\f(x) e^{-x }= e^{-x }/{-1}+c

⇒

\f(x) = - 1 + \ce^{x}

\f(0) = 0 = -1 + \ce^0 ⇒ c = 1

\f(x) = e^x - 1

\f(x) + \f(y) + \f(x) \f(y) = e^x- 1 + e^{y}- 1

+ (e^x - 1) (e^y- 1)

= e^x- 1 + e^y -1 + e^x.e^y - e^y - e^x + 1

= e^x.e^y - 1 = e^{x + y }- 1

= \f(x + y)

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