Question

what is the maximum value of function
f(x) = x^2 - 11x +30 ?

Answer 1

The function doesn't have a maximum since it opens up, i.e. the leading coefficient is positive/this is a monic 2nd degree polynomial.

Answer 2

to find a maximum f''(x)<0 according to the second derivative theorem
now, first we look for critical points with f'(x)=0
f'(x)=2x-11, so f'(x)=0 means
2x-11=0 -> x=11/2
there's our critic point, now
f''(x)=2, if f''(x)>0 the function have a minimum and not a maximum and puff