find the maximum value of f(x) where f(x)=integration of (2t−t²+5)dt from (0,x)

Find df/dx :
df/dx = 2x - x2+ 5 = 0

The roots of this equation : x = 1 + root(6), 1- root(6)
d/dx ( df/dx) = 2 - 2x
It is negative for the first root x = 1 + root(6)
Hence the maximum value can be found out by putting x = 1 + root(6) in the integral. Integrate the integral and find it.

Thanks & Regards
Bharat Bajaj
askIITians Faculty
IIT Delhi