Optimization

For each of the functions given below:

1. Find the critical points of the function
2. Compute the discriminant $D$ and evaluate $D$ for each critical point.
3. Use the Second Partial Derivatives Test to classify the critical points

1. $f(x,y) = 3x^3+y^2-9x+4y$

2. $f(x,y) = xy + \frac{2}{x} + \frac{4}{y}$

3. $f(x,y) = x^3 + y^3 - 3xy$

Respond to each of the following prompts to solve the given optimization problems.

1. Let $f(x,y) = \sin(x)+\cos(y)$. Determine the absolute maximum and minimum values of $f$. At what points do these extreme values occur?

2. For a certain differentiable function $F$ of two variables $x$ and $y$, its partial derivatives are Find each of the critical points of $F$, and classify each as a local maximum, local minimum, or a saddle point.

3. Deter…

If a continuous function $f$ of a single variable has two critical numbers $c_1$ and $c_2$ at which $f$ has relative maximum values, then $f$ must have another critical number $c_3$, because “it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between the mountains) or a local minimum (a true valley).” (From Calculus in Vector Spaces…

If a continuous function $f$ of a single variable has exactly one critical number with a relative maximum at that critical point, then the value of $f$ at that critical point is an absolute maximum. In this exercise we see that the same is not always true for functions of two variables. Let $f(x,y) = 3xe^y-x^3-e^{3y}$ (from “The Only Critical Point in Town” Test by Ira Rosenholz and Lowell Smylie in the Mathematics Magazine, V…

A manufacturer wants to procure rectangular boxes to ship its product. The boxes must contain 20 cubic feet of space. To be durable enough to ensure the safety of the product, the material for the sides of the boxes will cost $0.10 per square foot, while the material for the top and bottom will cost $0.25 per square foot. In this activity we will help the manufacturer determine the box of minimal…

A rectangular box with length $x$, width $y$, and height $z$ is being built. The box is positioned so that one corner is stationed at the origin and the box lies in the first octant where $x$, $y$, and $z$ are all positive. There is an added constraint on how the box is constructed: it must fit underneath the plane with equation $x + 2y + 3z = 6$. In fact, we will assume that the corner of the box “opposite” the origin must…

The airlines place restrictions on luggage that can be carried onto planes.

• A carry-on bag can weigh no more than 40 lbs.

• The length plus width plus height of a bag cannot exceed 45 inches.

• The bag must fit in an overhead bin.

Let $x$, $y$, and $z$ be the length, width, and height (in inches) of a carry on bag. In this problem we find the dimensions of the bag of largest volume, $V = xyz$, that satisfies the second…