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43 votes
43 votes
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?

The absolute minimum of f(x,y) = x^(2) + 4y^(2) -4y over the line segment x = 2 with-example-1
User Federico Sawady
by
2.8k points

1 Answer

18 votes
18 votes

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)

User Robert Jacobs
by
3.0k points