Let f : R → R be a continuous function satisfying \f(x) +\int ^{ x} _{0} \tf(t)dt +x^2 =0 , for all x ∊R. Then -

Correct Answer is : limx→−∞f(x)=−2

f(x) has more than one point in common with the x-axis

KVPY SX SB Exam 2015 Question Paper

Question : 87 of 120

Marks: +1, -0

Let f : R → R be a continuous function satisfying

\f(x) +∫ ↖{ x} ↙{0} \tf(t)dt +x^2 =0 ,

for all

x ∊R

. Then -

Solution:

\f(x) +∫ ↖{ x} ↙{0} \t \ \f(t)dt +x^2 =0 ,

f'(x) + xf(x) + 2x = 0

{\f'(x)} / {\f(x) + 2} =-x

∫{\f'(x)} / ({\f(x) + 2}) =-xdx

ln (\f(x) + 2) = –x^2/2+c

\f(x) + 2 = 2 e ^{ -x^2 \/ 2} + c

\f(x) = k\ e ^{ -x^2 \/ 2}- 2

wherex = 0 ,f(x) = 0, k = 2

\f(x) = 2\( e ^{ -x^2 \/ 2}- 1)

clearly

\lim↙{x→∞} \f(x) = -2

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