Chapter 12: Q. 26 (page 985)

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.
$f (x, y) = x y when x^{2} + y^{2} = 16$

Short Answer

Expert verified

The maximum value of the function is $8$ and the minimum value is $- 8$ and both exist as the constraint is a bounded and closed circle of radius $4 .$

Step by step solution

Step 1. Given information.

Given function is $f (x, y) = x y .$

Given constraint is $x^{2} + y^{2} = 16 .$

Step 2. critical points of the function.

Gradients of function.

$\nabla f (x, y) = y i + x j \nabla g (x, y) = 2 x i + 2 y j$

Use the method of Lagrange multipliers.

$\nabla f (x, y) = λ \nabla g (x, y) y i + x j = λ (2 x i + 2 y j) y i + x j = 2 λ x i + 2 λ y j$

Compare terms.

$y = 2 λ x \Rightarrow λ = \frac{y}{2 x} x = 2 λ y \Rightarrow λ = \frac{x}{2 y} so x^{2} = y^{2}$

substitute $x^{2} = y^{2}$ in constraint.

$x^{2} + y^{2} = 16 y^{2} + y^{2} = 16 2 y^{2} = 16 y = \pm 2 \sqrt{2} x = \pm 2 \sqrt{2}$

so critical points are $(- 2 \sqrt{2}, - 2 \sqrt{2}), (2 \sqrt{2}, - 2 \sqrt{2}), (- 2 \sqrt{2}, 2 \sqrt{2}) & (2 \sqrt{2}, 2 \sqrt{2}) .$

Step 3. maximum and minimum of a function.

Find function value at $(- 2 \sqrt{2}, - 2 \sqrt{2}), (2 \sqrt{2}, - 2 \sqrt{2}), (- 2 \sqrt{2}, 2 \sqrt{2}) & (2 \sqrt{2}, 2 \sqrt{2}) .$

$f (- 2 \sqrt{2}, - 2 \sqrt{2}) = 8 f (2 \sqrt{2}, - 2 \sqrt{2}) = - 8 f (- 2 \sqrt{2}, 2 \sqrt{2}) = - 8 f (2 \sqrt{2}, 2 \sqrt{2}) = 8$

So the maximum value of the function is $8$ and the minimum value is $- 8 .$

As constraint is bounded and closed circle of radius $4$ so maximum and minimum must both exist.

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In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.
$f (x, y) = x y when x^{2} + y^{2} = 16$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. critical points of the function.

Step 3. maximum and minimum of a function.

One App. One Place for Learning.

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In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.f(x,y)=xywhenx2+y2=16

Short Answer

Step by step solution

Step 1. Given information.

Step 2. critical points of the function.

Step 3. maximum and minimum of a function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Decision Maths

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.
$f (x, y) = x y when x^{2} + y^{2} = 16$