1. Chapter 14 – Partial Derivatives
14.7 Maximum and Minimum Values
Objectives:
Use directional derivatives to locate maxima and minima of multivariable functions
Maximize the volume of a box without a lid if we have a fixed amount of cardboard to work with
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14.7 Maximum and Minimum Values

2. Absolute & Local Maximum
There are two points (a, b) where f has a local maximum—that is, where f(a, b) is larger than nearby values of f(x, y).
The larger of these two values is the absolute maximum.
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14.7 Maximum and Minimum Values

3. Absolute & Local Minima
Likewise, f has two local minima—where f(a, b) is smaller than nearby values.
The smaller of these two values is the absolute minimum.
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14.7 Maximum and Minimum Values

4. Definition – Local Max and Min
If the inequalities in Definition 1 hold for all points (x, y) in the domain of f, then f has an absolute maximum (or absolute minimum) at (a, b).
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14.7 Maximum and Minimum Values

5. Theorem
Notice that this theorem can be stated in the notation of gradient vectors as ∇f(a, b) = 0.
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14.7 Maximum and Minimum Values

6. Definition – Critical Point
A point (a, b) is called a critical point (or stationary point) of f if either:
fx(a, b) = 0 and fy(a, b) = 0
One of these partial derivatives does not exist.
Theorem 2 says that, if f has a local maximum or minimum at (a, b), then (a, b) is a critical point of f.
At a critical point, a function could have a local maximum or a local minimum or neither.
Dr. Erickson
14.7 Maximum and Minimum Values

7. Saddle Point
A function need not have a maximum or minimum value at a critical point.
The figure shows how this is possible.
The graph of f is the hyperbolic paraboloid z = y2 – x2.
It has a
horizontal tangent plane (z = 0) at the
origin.
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14.7 Maximum and Minimum Values

8. Saddle Point You can see that f(0, 0) = 0 is:
A maximum in the direction of the x-axis.
A minimum in the direction of the y-axis.
Near the origin, the graph has the shape of a saddle.
So, (0, 0) is
called a saddle
point of f.
Dr. Erickson
14.7 Maximum and Minimum Values

9. Extreme Value at a Critical Point
We need to be able to determine whether or not a function has an extreme value at a critical point.
The following test is analogous to the Second Derivative Test for functions of one variable.
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14.7 Maximum and Minimum Values

10. Second Derivative Test
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14.7 Maximum and Minimum Values

11. Notes on Second Derivative Test
Note 1: In case c,
The point (a, b) is called a saddle point of f .
The graph of f crosses its tangent plane at (a, b).
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14.7 Maximum and Minimum Values

12. Notes on Second Derivative Test
Note 2: If D = 0, the test gives no information:
f could have a local maximum or local minimum at (a, b), or (a, b) could be a saddle point of f.
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14.7 Maximum and Minimum Values

13. Notes on Second Derivative Test
Note 3: To remember the formula for D, it’s helpful to write it as a determinant:
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14.7 Maximum and Minimum Values

14. Visualization Critical Points from Contour Maps Families of Surfaces
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14.7 Maximum and Minimum Values

15. Example 1
Find the local maximum and minimum values and saddle point(s) of the function.
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14.7 Maximum and Minimum Values

16. Definition – Closed Set
Just as a closed interval contains its endpoints, a closed set in ℝ2 is one that contains all its boundary points.
NOTE: A boundary point of D is a point (a, b) such that every disk with center (a, b) contains points in D and also points not in D.
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14.7 Maximum and Minimum Values

17. Closed Set
If even one point on the boundary curve were omitted, the set would not be closed.
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14.7 Maximum and Minimum Values

18. Definition – Bounded Set
A bounded set in ℝ2 is one that is contained within some disk.
In other words, it is finite in extent.
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14.7 Maximum and Minimum Values

19. Theorem
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14.7 Maximum and Minimum Values

20. Closed Interval Method
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14.7 Maximum and Minimum Values

21. Example 2 – pg. 954
Find the absolute maximum and minimum values of f on the set D.
#32.
#34.
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14.7 Maximum and Minimum Values

22. Example 3 – pg.955 # 48
Find the dimensions of the rectangular box with the largest volume if the total surface area is given as 64 cm2.
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14.7 Maximum and Minimum Values

23. Example 4 – pg. 955 #51
A cardboard box without a lid is to have a volume of 32,000 cm3. Find the dimensions that minimize the amount of cardboard used.
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14.7 Maximum and Minimum Values

24. More Examples
The video examples below are from section in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 3
Example 5
Example 6
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14.7 Maximum and Minimum Values

25. Demonstrations Feel free to explore these demonstrations below.
Second Order Partial Derivatives
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14.7 Maximum and Minimum Values