The absolute minimum of f(x,y) = x^(2) + 4y^(2) -4y over the line segment x = 2 with 0 ≤ y ≤ 2 equals which of the following?

Question

The absolute minimum of
`f(x,y) = x^(2) + 4y^(2) -4y` over the line segment x = 2 with 0 ≤ y ≤ 2 equals which of the following?

Answer 1

On the line segment
`x=2`,
`f` depends only on
`y` :

`f(2,y) = g(y) = 4y^2 - 4y - 4`

Find the critical points of
`g`.

`(dg)/(dy) = 8y - 4 = 0 \implies y = \frac12`

At this critical point, we have

`g\left(\frac12\right) = f\left(2,\frac12\right) = 3`

Check the value of
`f` at the endpoints of the line segment.

`f\left(2,0\right) = 4`

`f\left(2,2\right) = 12`

So we have

`\min\left\{f(x,y) \mid x=2 \text{ and } 0 \le y \le 2\right\} = \boxed{3} \text{ at } (x,y) = \left(2,\frac12\right)`