Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Finding Global Extrema

Calculus

5. Graphical Applications of Derivatives

Finding Global Extrema

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Multiple Choice

Find the global maximum and minimum values of the function $f (x, y) = e^{- x y}$ on the region defined by $x^{2} + 4 y^{2} \leq 1$ .

The global maximum is

1

(0, 0)

and the global minimum is

e^{- 0.5}

(0, 0.5)

and

(0, - 0.5)

The global maximum is

e

(0, 0.5)

and the global minimum is

e^{- 0.5}

(1, 0)

and

(- 1, 0)

The global maximum is

e^{- 0.5}

(0, 0.5)

and the global minimum is

e

(1, 0)

and

(- 1, 0)

The global maximum is

e

(0, - 0.5)

and the global minimum is

e^{- 0.5}

(1, 0)

and

(- 1, 0)

Previous problemNext problem

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the global maximum and minimum values of the function f(x, y) = e^(-xy) within the region defined by x^2 + 4y^2 ≤ 1. This involves analyzing the function's behavior both inside the region and on its boundary.

Step 2: Identify the critical points. To find critical points, compute the partial derivatives of f(x, y) with respect to x and y, set them equal to zero, and solve for x and y. The partial derivatives are: ∂f/∂x = -y * e^(-xy) and ∂f/∂y = -x * e^(-xy). Solve ∂f/∂x = 0 and ∂f/∂y = 0 simultaneously.

Step 3: Analyze the boundary of the region. The boundary is defined by the equation x^2 + 4y^2 = 1. Parameterize the boundary using trigonometric functions, such as x = cos(t) and y = (1/2)sin(t), where t is a parameter. Substitute these into f(x, y) to express f in terms of t, and find the maximum and minimum values of f(t) by analyzing its derivative.

Step 4: Evaluate f(x, y) at the critical points and boundary points. Substitute the critical points and the points obtained from the boundary analysis into f(x, y) = e^(-xy) to determine the function's values at these locations.

Step 5: Compare the values obtained in Step 4. The largest value corresponds to the global maximum, and the smallest value corresponds to the global minimum. Ensure that the points satisfy the region's constraint x^2 + 4y^2 ≤ 1.