1. Find the local maximum and minimum values and saddle points of the function. Graph the...

Question:

1. Find the local maximum and minimum values and saddle points of the function. Graph the function with a domain and viewpoint that reveal all the important aspects of the function.

$$f(x, y) = xy + \frac{27}{x} + \frac{27}{y}$$

2. The base of an aquarium with given volume {eq}V {/eq} is made of slate and the sides are made of glass. If the slate costs five times as much (per unit area) as glass, use Lagrange multipliers to find the dimensions of the aquarium that minimize the cost of the materials.

Find and Classify Critical Points:

The critical points of the function {eq}f(x,y) {/eq} are the solutions to the system of equations of the form {eq}\nabla f(x,y) = 0 {/eq}. Then, we classify the critical points using the second partial derivative test. In this test, we compute the value of the determinant {eq}D(a,b) = f_{xx}(a,b)f_{yy}(a,b) - (f_{xy}(a,b))^2 {/eq} for each critical point {eq}(a,b) {/eq}. The following guidelines allow us to classify each critical point:

1. If {eq}D(a,b) > 0 \text{ and } f_{xx}(a,b) > 0 \text{ then } (a,b) {/eq} is a minimum.

2. If {eq}D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{ then } (a,b) {/eq} is a maximum.

3. If {eq}D(a,b) < 0 \text{ then } (a,b) {/eq} is a saddle point.

4. If {eq}D(a,b) = 0 {/eq} then the test is inconclusive.

Given an objective function {eq}f(x,y) {/eq} and a constraint function {eq}g(x,y) {/eq}, we can use the method of Lagrange multipliers to find the critical points. The critical points are the solutions to the system of equations of the form {eq}\nabla f(x,y) = \lambda \nabla g(x,y) {/eq} and {eq}g(x,y) = k {/eq} where {eq}\lambda {/eq} is the Lagrange multiplier. We classify the critical points by comparing their objective function values. The largest function value is the maximum and the smallest function value is the minimum.

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1. Find the local maximum and minimum values and saddle points of the function. Graph the function with a domain and viewpoint that reveal all the...

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Find the Maximum Value of a Function: Practice & Overview

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Chapter 10 / Lesson 12
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Learn how to find the maximum or minimum value of a quadratic function, and which functions have minimum or maximum values.