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: f(ln 2) is greater than

Step-by-step solution

We have f(x)= _ 0 e^x ^4 ₀^ (e^t-1 ) ^2(1+t) (2 t^3+3 ) d t [ 0 0 . form ] aligned & = _ 0 e ^ x ( e ^ -1 ) ^2(1+ ) (2 ^3+3 ) 4 ^3 & = e ^ x _ 0 e ^ -1 ^2(1+ ) ^2 1 4 (2 ^3+3 ) & = e ^ x 12 aligned Clearly f( 2)= e^ 2 12 = 1 6