Question

The absolute minimum of rmula1" title="f(x,y) = x^{2} + 4y^{2} -4y" alt="f(x,y) = x^{2} + 4y^{2} -4y" align="absmiddle" class="latex-formula"> 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 $g$.

$\dfrac{dg}{dy} = 8y - 4 = 0 \implies y = \dfrac12$

At this critical point, we have

$g\left(\dfrac12\right) = f\left(2,\dfrac12\right) = 3$

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

$\min\left\{f(x,y) \mid x=2 \text{ and } 0 \le y \le 2\right\} = \boxed{3} \text{ at } (x,y) = \left(2,\dfrac12\right)$