Absolute Minima and Maxima on Triangles at the boundary Find absolute minimum and maxima for $f(x, y)= 11-3x+7y$
on closed triangular region with vertices $(0, 0), (7, 0)$ and $(7, 11)$.
How would I approach this type of question. I understand how to find the critical points on the interior which is just the partial derivative with respect to $x$ and $y$ which is $-3$ and $7$. But how do I find the critical points on the boundary with the vertices. I do not know what the end points would be to aid me with this question.
 A: There are two ways we can approach this problem. One is the traditional calculus way. You would first have to check the interior of the region and look for critical points. It's straightforward to see that there isn't one. We now must test the boundaries, which are lines. For each side of the triangle, we can treat it as a constrained optimization problem. For example, when looking for optimal solution along the side with endpoints $(0,0)$ and $(7,0)$, we can treat it as the following problem:
$$\text{optimize } f(x,y)=11-3x+7y \text{, subject to } g(x,y) = y = 0$$
You can use the method of Lagrange multipliers to solve this equation:
$$\nabla f=\lambda \nabla g \text{, at optimal points} \implies (-3, 7) = \lambda\cdot(0,1)$$
which clearly has no solution. Repeating this method on each side will yield a similar result. The last candidates to be checked are the vertices themselves (they can be seen as "boundaries" of the sides). Checking their values will show that $f(7,11)$ is the absolute maximum, and $f(7,0)$ is the absolute minimum.
The second way makes use of principles in linear programming. We know that this is equivalent to the problem of finding the absolute minima and maxima of the linear function $f(x,y)=-3x+7y$ over the triangle. If you've studied the problem of linear programming, you know that optimal solutions (max and min) all occur at extreme points of a convex set. The triangular region you're considering is convex, so really, you only need to evaluate the function at the vertices to find the absolute min and max.
A: $f(x,y)$ is "flat"
That is, when we calculate the partial derivatives we get constants.  There are no critical points in the interior of the region.
But since we have a function on a closed and bounded domain, there must be extreme values.  And, if these extreme values are not on the interior, we should be checking the boundary.
The boundary can be modeled with parametric equations.
$x = 7t, y = 0\\
x = 7, y = 11t\\
x = 7t, y = 11t$
And $0\le t \le 1$ for each of these curves.
But making these substitutions, for each "curve", we get something of the form:
$f(t) = \alpha t + \beta $
Normally we would take the derivative with respect to our parameter and stet that equal to 0.  But again, we can see we will have no luck.
Next thing to do is to check the corners.  These are our extreme values.
$f(7,11) > f(0,0) > f(7,0)$
Based on the set-up, you could probably jump to the conclusion.  But, you should probably do the middle steps, until you have built up your intuition.
