3. Optimize f on the triangle with corners (0,0), (0,5), (5,0). Guide: The boundaries are y=0, x=0, y-5-x. Sketch this,

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3 Optimize F On The Triangle With Corners 0 0 0 5 5 0 Guide The Boundaries Are Y 0 X 0 Y 5 X Sketch This 1 (60.39 KiB) Viewed 33 times

3. Optimize f on the triangle with corners (0,0), (0,5), (5,0). Guide: The boundaries are y=0, x=0, y-5-x. Sketch this, and verify that the interior satisfies: y>0, x>0, y<5-x. 1. The interior critical points are the same as in (1) and (2), but you have to check if they are inside or outside your domain, by checking the three inequalities above. lla. The boundaries are straight lines. While Lagrange Multipliers for all three boundaries is an option, this is slightly easier by substitution. The substitution method: Each side of the triangle involves looking for critical points of the single-variable expressions: f(x,0), 0<x<5, bottom side. f(0,y), 0<y<5, left side f(x,5-x), 0<x<5, hypotenuse llb. Corners. When the boundary is piecewise, there can curves AND corners to deal with. In higher dimensions, a boundary could be multiple surfaces, which connect along curves, which could also have corners! For this problem, we only have lines and corners. The corners are (0,0), (5,0), and (0,5). Check these. Sometimes the max and/or min may occur on a corner instead of a critical point. Therefore, corners play the same role as a critical point, and belong on the same list. III. Check all critical points and corners.