KVPY SX SB Exam 2018 Question Paper

Question : 14 of 120

Marks: +1, -0

Let S be the set of real numbers p such that there is no nonzero continuous function f : R → R satisfying

∫↖{ x}↙{0}\f (t)dt = p \f(x)

for all x ∈ R. Then S is

Solution:

∫↙{0}↖{x} f(t) d t=p f(x) . . .(1)

Put

x=0

in eq. (1)

f(0)=0 . . .(2)

On differentiating eq.(1)

p f^{'}(x)=f(x)

f^{'}(x) \/ f(x)=1 \/ p

On integrating both sides

\ln f(x)=x \/ p+c

f(x)=K e^{x \/ p}

f(0)=0

f(0)=K e^{0}=0

K=0

So,

f(x)=0

Hence

S ∈ R

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