find the maximum value of f(x) =integration of 0 to x (2t-t²+5)dt

on integrating using lebnitz theorem for differantiation under integral sign df(x)/dx=2x-x2+5 for maxima and minima df(x)/dx=0 =>2x-x2+5=0 => x= 1+sqrt(6),1-sqrt(6) d2f(x)/dx2=2-2x ,d2f(x)/dx2=sqrt(6) for x= 1-sqrt(6) and ,d2f(x)/dx2=-sqrt(6) for x= 1+sqrt(6) thus f(x) is maximum for x= 1+sqrt(6) as,d2f(x)/dx2=-sqrt(6)<0 thus maximum value of f(x)=integral 0 to1+sqrt(6)(2t-t²+5)dt = (t2-t3/3+5t)01+sqrt(6) =17/3 + 4sqrt(6)

Thanks and Regards,
M.MURALIKRISHNA
askIITians faculty

Approved

a