1. Maxima and Minima of Functions of Several Variables (7.3)

Note: Local is another way to say relative.

Maximum and minimum valuesare called extreme values of f .

Maxima and minima must occurwithin the domain of f .

Definition 1.1. A function of two variablesf(x, y) has a relative maximum when x = aand y = b if f(x, y) is at most equal to f(a, b)whenever x is near a and y is near b. In otherwords, f(x, y) has a relative maximum at (a, b)if f(x, y) ≤ f(a, b) when (x, y) is near (a, b).Geometrically, the graph of f(x, y) has a peak at(x, y) = (a, b). The number f(a, b) is called arelative maximum value.

A function of two variables f(x, y) has arelative minimum when x = a and y = b iff(x, y) is at least equal to f(a, b) whenever x isnear a and y is near b. In other words, f(x, y)has a relative minimum at (a, b) if f(x, y) ≥f(a, b) when (x, y) is near (a, b).Geometrically, the graph of f(x, y) has a valleywhose bottom occurs at (x, y) = (a, b). The num-ber f(a, b) is called a relative minimum value.

Definition 1.2. If f(x, y) ≤ f(a, b) for all points (x, y) in the domain of f , then f has an absolutemaximum at (a, b), and f(a, b) is an absolute maximum value.

If f(x, y) ≥ f(a, b) for all points (x, y) in the domain of f , then f has an absolute minimum at(a, b), and f(a, b) is an absolute minimum value.

First Derivative Test for Functions of Two Variables: If f(x, y) has either a relative maxi-mum or minimum at (x, y) = (a, b) (and the first-order partial derivatives of f exist), then

∂f

∂x(a, b) = 0 and

∂f

∂y(a, b) = 0.

The geometric interpretation of this test is that at a relative maximum or minimum, the tangentplane to the graph of f(x, y) must be horizontal.

A point (a, b) in the domain of f(x, y) is called a critical point of f if∂f

∂x(a, b) = 0 and

∂f

∂y(a, b) = 0.

If f has a local maximum or minimum at (a, b), then (a, b) is a critical point of f . However, not allcritical points give rise to maxima or minima.

1

Maxima and Minima 2

Example 1.1. The function f(x, y) = x2 + y2− 2x− 6y+ 14 has a minimum. Find the point (x, y)where the minimum occurs and compute the minimum value of the function.

Example 1.2. Find all points (x, y) where the function f(x, y) = 2x2 − 4xy + y4 has a possiblerelative maximum or minimum.

We find points (x, y) where f(x, y) has potential relative maxima or minima by simultaneously

solving∂f

∂x= 0 and

∂f

∂y= 0. The following test helps us to determine if a point is a relative

maximum, a relative minimum, or neither. There is also a possibility for the test to be inconclusive.

Second Derivative Test for Functions of Two Variables: Suppose that f(x, y) is a function

and (a, b) is a point at which∂f

∂x(a, b) = 0 and

∂f

∂y(a, b) = 0. Let D(x, y) =

∂2f

∂x2· ∂

2f

∂y2−(

∂2f

∂x∂y

)2

.

1) If D(a, b) > 0 and∂2f

∂x2(a, b) < 0, then f(x, y) has a relative maximum at (a, b).

2) If D(a, b) > 0 and∂2f

∂x2(a, b) > 0, then f(x, y) has a relative minimum at (a, b).

3) If D(a, b) < 0, then f(x, y) has neither a relative maximum nor a relative minimum at (a, b).(f(x, y) is said to have a saddle point at (a, b)).

4) If D(a, b) = 0, no conclusion can be drawn from this test.

Maxima and Minima 3

Example 1.3. Find all points (x, y) where the function f(x, y) has a possible relative maximum orminimum. Use the Second Derivative Test to determine, if possible, the nature of f(x, y) at each ofthese points. If the Second Derivative Test is inconclusive, so state.

a) f(x, y) = 3x2y + y3 − 3x2 − 3y2 + 2

b) f(x, y) = x4 + y4

Example 1.4. The function f(N,P ) = NPe−(2N+P ) describes the yield of some crop in terms ofthe amount of nitrogen (N) and phosphorus (P ) applied.

a) Find all points (N,P ) where the function has potential relative maxima or minima.

b) Using the following information and the Second Derivative Test determine the nature of thepoints found in Part a).

∂2f

∂N2(0, 0) = 0,

∂2f

∂P 2(0, 0) = 0,

∂2f

∂N∂P(0, 0) = 1,

∂2f

∂N2

(1

2, 1

)= −2e−2,

∂2f

∂P 2

(1

2, 1

)= −1

2e−2,

∂2f

∂N∂P

(1

2, 1

)= 0

Example 1.5. A rectangular box without a lid is to be made from 12 m2 of cardboard. Find themaximum volume of such a box.