Let f(x)= [ array lr e^x ( e^ n x -1 e^x-1 +x^3 ), & x 0 k, & x=0 array . (where n N ) be a differentiable function and if f^ (0)=1302 , then find the value of (k+n) .

Step-by-step solution

aligned & k= _ x 0 ( e^x (e^ n x -1 ) e^x-1 +e^x x^3 )=n & f(x)=e^x+e^ 2 x +e^ 3 x + .+e^ n x +e^x x^3 & f^ (x)=e^x+2 e^ 2 x +3 e^ 3 x + .+n e^ n x +e^x x^3+e^x 3 x^2 & f^ (x)=e^x+2^2 e^ 2 x +3^2 e^ 3 x + .+n^2 e^ n x +e^x x^3+6 e^x x^2+e^x 6 x & f^ (x)=e^x+2^3 e^ 2 x +3^3 e^ 3 x + .+n^3 e^ n x +e^x x^3+9 e^x x^2+e^x 18 x+6 e^x & f^ (0)=1^3+2^3+3^3+ .+n^3+6=1302 & ( n(n+1) 2 )^2=1296 n=8 aligned k+n=16