    Question

# Define g(x)=3∫−3f(x−y)f(y) dy, for all real x, where f(t)={1 0≤t≤10 elsewhere. Then

A
g(x) is not continuous everywhere
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B
g(x) is continuous everywhere but differentiable nowhere
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C
g(x) is continuous everywhere and differentiable everywhere except at x=0,1
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D
g(x) is continuous everywhere and differentiable everywhere except at x=0,1,2
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Solution

## The correct option is D g(x) is continuous everywhere and differentiable everywhere except at x=0,1,2g(x)=3∫−3f(x−y)f(y) dy⇒g(x)=1∫0f(x−y) dy Assuming x−y=t⇒dy=−dt So, g(x)=x∫x−1f(t) dt Case I : x<0 f(t)=0⇒g(x)=0 Case II : 0≤x<1 g(x)=0∫x−1f(t) dt+x∫0f(t) dt⇒g(x)=x∫01 dt=x Case III : 1≤x≤2 g(x)=1∫x−1f(t) dt+x∫0f(t) dt⇒g(x)=1∫x−11 dt=2−x Case IV : x>2 f(t)=0⇒g(x)=0 ∴g(x)=⎧⎪ ⎪⎨⎪ ⎪⎩0 x<0x 0≤x<12−x 1≤x≤20 x>2 Clearly, g(x) is continuous ∀ x∈R and g(x) is differentiable ∀ x∈R−{0,1,2}.  Suggest Corrections  0      Similar questions  Related Videos   Parametric Differentiation
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