Paragraph: Let f and g be two real valued differentiable function on R satisfying ₀^x g(t) d t=3 x+ _x^0 ^2 t g(t) d t and f(x)= _ 0 1 ^4 ₀^ (e^ x+t -e^x ) ^2(1+t) 2 t^3+3 d t Question: Range of g(x) is equal to

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Correct answer: A. [ 3 2 , 3 ]

We have ₀^x g(t) d t=3 x+ _x^0 ^2 t g(t) d t On differentiating both side w.r.t x aligned & g(x)=3- ^2 x g(x) & g(x)= 3 1+ ^2 x aligned Clearly 3 2 3 1+ ^2 x 3

Paragraph: Let f and g be two real valued differentiable function on R satisfying ₀^x g(t) d t=3 x+ _x^0 ^2 t g(t) d t and f(x)= _ 0 1 ^4 ₀^ (e^ x+t -e^x ) ^2(1+t) 2 t^3+3 d t Question: Range of g(x) is equal to

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