`f(x)` is polynomial function of degree 6, which satisfies `("lim")_(xvec0)(1+(f(x))/(x^3))^(1/x)=e^2` and has local maximum at `x=1` and local minimum at `x=0a n dx=2.` then 5f(3) is equal to

Answer

4

Explanation

Let f(x)=`a_(0)+a_(1)x+a_(2)x+a_(3)x^(3)+a_(4)x^(4)+a_(5)x^(4)+a_(5)x^(5)+a_(6)x^(6)`
Give `underset(xrarra)lim1+f(x)/(x^(3))^(1//3)=e^(2)`
`therefore underset(xrarr0)lim(f(x))/(x^(3))=0`
or `fa_(0)=a_(1)=a_(2)=a_(3)=0`
or `therefore8+5a_(5)+6a_(6)=0`
and `4+5a+_(5)+12a_(6)=0`
solving we get
`a_(5)=(12)/(5),a_(6)=2/3`
`therefore f(x)=2x^(4)-12/5x^(5)+2/3x^(6)`