Define g(x)=\int^{3}_{-3}\f(x -y)\f(y)dy for all real x,where \f(t)=\\begin{array}{cc}1{ &\end{array},0\leq t\leq 1{,};0{,},\elsewhere Then

Correct Answer is : g(x) is continuous everywhere and differentiable everywhere except at x = 0,1,2

g(x) is not continuous everywhere

g(x) is continuous everywhere but differentiable nowhere

g(x) is continuous everywhere and differentiable everywhere except at x = 0,1

g(x) is continuous everywhere and differentiable everywhere except at x = 0,1,2

KVPY SX SB Exam 05-Nov-2017 Question Paper

Question : 88 of 120

Marks: +1, -0

Define

g(x)=∫↖{3}↙{-3}\f(x -y)\f(y)dy

for all real x,where

\f(t)=\{\table 1{,},0≤ t≤ 1{,};0{,},\elsewhere

Then

Solution:

Definition can be break as

g(x)=∫↖{1}↙{0}\f(x -y) dy

x – y = t ; –dy . dt

g(x) = ∫↖{x}↙{x-1}\f (t) dt

$g(x) = \{\table 0,x≤0;x,02$
Now, check yourself

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