calcus help ASAP

Question

(1 point) For each of the following functions, find the maximum and minimum values of the function on the circular disk: 𝑥2+𝑦2≤1. Do this by looking at the level curves and gradients.

(A) 𝑓(𝑥,𝑦)=𝑥+𝑦+4

maximum value = 

minimum value = 

Answer 1

In general, a maximum or a minimum occurs either at a critical point or on the boundary.

Lets first consider the boundary, which is the unit circle

x² + y² = 1

This circle does not have any boundary itself, such as end points.

Therefore the minimum or maximum of f(x, y) restricted to the circle

must occur at a critical point of the function F:

F(x, y, λ) = f(x, y) - λ(x² + y² - 1) = x + y + 4 - λ(x² + y² - 1)

(This last statement about F is what is meant by the method of gradients,

or Lagrange multipliers.)

The critical points of F are those satisfying

0 = ∂F/∂x = 1 - 2λx (1)

0 = ∂F/∂y = 1 - 2λy (2)

0 = ∂F/∂λ = -(x² + y² - 1) (3)

From equation (1)

x = 1/2λ (4)

From equation (2)

y = 1/2λ (5)

Substituting both into (3)

1/4λ² + 1/4λ² - 1 = 0

Rearrange into

λ² = 1/2

which has two solutions λ = 1/√2 and -1/√2.

When we plug those two solutions into (4) and (5) we get the critical points

(√2/2, √2/2) and (-√2/2, -√2/2)

(These are candidates for being maxima and minima of the function f(x,y).)

The corresponding function values are

f(√2/2, √2/2) = √2 + 4

f(-√2/2, -√2/2) = -√2 + 4

The other candidates for maximum or minimum are critical points of the function f(x,y)

on the open domain of the inside of the circle x² + y² < 1

Those critical points satisfy

0 = ∂f/∂x = 1

0 = ∂f/∂y = 1

That means f(x, y) has no critical points.

Therefore the maximum of f(x, y) on the unit disk occurs at (√2/2, √2/2),

and the minimum occurs at (-√2/2, -√2/2),