Question

What are the extrema of `f(x)=x^2 - 6x + 11` on `x in[1,6]`?

Answer 1

`(3,2)` is a minimum.
`(1,6) and (6,11)` are maxima.

Explanation:

Relative extrema occur when `f'(x)=0`.
That is, when `2x-6=0`.
ie when `x=3`.

To check if `x=3` is a relative minimum or maximum, we observe that `f''(3)>0` and so `=> x=3` is a relative minimum,
that is, `(3, f(3))=(3,2)` is a relative minimum and also an absolute minimum since it is a quadratic function.

Since `f(1)=6 and f(6)=11`, it implies that `(1,6) and (6,11)` are absolute maxima on the interval `[1,6]`.

graph{x^2-6x+11 [-3.58, 21.73, -0.37, 12.29]}