Understanding Absolute Extrema in Multivariable Functions

The video tutorial explains how to find the absolute extrema of a function f(x, y) = 2x + 5y over a closed interval for both x and y. It covers the process of finding critical points, determining extrema on boundaries using calculus techniques, and verifying results graphically. The tutorial concludes with identifying the absolute maximum and minimum values of the function within the specified region.

What is the function f(x, y) that we are analyzing for absolute extrema?

• f(x, y) = x^2 + y^2

• f(x, y) = x + y

• f(x, y) = 2x + 5y

• f(x, y) = 3x + 4y

What is the first step in finding the critical points of a function of two variables?

• Graph the function

• Find where the partial derivatives are zero or undefined

• Evaluate the function at the endpoints

• Set the function equal to zero

Why are there no critical points for the function f(x, y) in this case?

• The function is linear

• The function is not defined on the given interval

• The partial derivatives are constants and never zero

• The function is not continuous

When evaluating the boundary x = 2, what is the function g(y) equivalent to?

• g(y) = 5x + 2y

• g(y) = 4 + 5y

• g(y) = 2y + 5

• g(y) = 2x + 5y

What is the absolute maximum value found along the boundary x = 2?

• 6

• 14

• -14

• -6

What is the function g(y) when evaluating the boundary x = -2?

• g(y) = 5x + 2y

• g(y) = 2x - 10

• g(y) = 4 + 5y

• g(y) = -4 + 5y

What is the absolute minimum value found along the boundary x = -2?

• 14

• -14

• 6

• -6

What is the function g(x) when evaluating the boundary y = -2?

• g(x) = 2x - 10

• g(x) = 2x + 10

• g(x) = -4 + 5y

• g(x) = 4 + 5y

What is the absolute maximum value found along the boundary y = 2?

• -14

• -6

• 14

• 6

Where does the absolute minimum value occur in the bounded region?

• (1, 1)

• (2, 2)

• (-2, -2)

• (0, 0)