10.2: Multivariable Optimization
We now investigate the algebraic ideas involved in multivariable optimization.
Algebraically Finding Critical Points
Recall, from the definitions in 10.1, all critical points for a 3-D function z = f ( x, y) occur
where both the x and y cross-sectional functions have local extrema.
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The cross-section models are 2-D functions, and 2-D functions have local extrema
where their derivative = 0.
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The point (a, b) is a critical point of z = f ( x, y) if fx ( a, b) = 0 and fy ( a, b) = 0 .
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So we must take each of the first partial derivatives, set them both equal to 0, and
solve for all solutions to find the critical points.
Classifying Critical Points
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é f
xx
First we find the Hessian: ê
ê f yx
ë
f xy ù
ú
f yy ú
û
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We then form the determinant function: D = fxx × fyy - fyx × fxy
Determinant Test
1. If D ( a, b) > 0 and fxx ( a, b) < 0 , there is a local maximum at (a, b).
2. If D ( a, b) > 0 and fxx ( a, b) > 0 , there is a local minimum at (a, b).
3. If D ( a, b) < 0 , there is a saddle point at (a, b).
4. If D ( a, b) = 0 , we know nothing. We will not see this case.
Example 1
Find and classify all critical points of the function f ( x, y) = xy + 2y2 + x 2 . Some graphs are
provided, but you must show algebraic support of your answers.
Example 2
Find and classify all critical points of the function f ( x, y) = 4xy - x 4 - y 4 . Some graphs are
provided, but you must show algebraic support of your answers.
Example 3
Find and classify all critical points of the function f ( x, y) = -x 2 + 4x - y2 + 2y +12 . Some
graphs are provided, but you must show algebraic support of your answers.
Example 4
Find and classify all critical points of the function f ( x, y) = 121 x 3 + 14 y2 + 23 y - xy . Some
graphs are provided, but you must show algebraic support of your answers.
Example 5
Find and classify all critical points of the function f ( x, y) = 12 xy2 - 2x 2 y + 3x 2 + 36x . Some
graphs are provided, but you must show algebraic support of your answers.
Example 6
Find and classify all critical points of the function f ( x, y) = 3sin ( p2 x ) +1.5y2 - 6y + 6 under
the restriction that 0 £ x £ 4 . Some graphs are provided, but you must show algebraic
support of your answers.
Example 7
A nursery sells mulch by the truckload. Bark mulch sells for $b per load while pine straw
sells for $p per load. The average weekly profit from the sale of these two mulches can be
modeled by the formula A ( p, b) =144p - 3p2 - pb - 2b2 +120b + 35 dollars. How much
should be charged for each type of mulch to maximize the weekly profits. What are the
maximum average weekly profits? Verify that your solution is a maximum.
Example 8
At a certain assembly plant, some data analysis has shown that the percentage of product that
is flawed can be modeled by F ( x, y) = 0.3( x - 3) + 0.1( y - 6) + 0.03xy + 0.2 percent where x
is the average number of workers assigned concurrently to one assembly station and y is the
average number of hours that each worker spends on task during a shift. Under the
restrictions 1£ x £ 5 and 1< y £11 , find the conditions that minimize the percentage of
product that is flawed. What is this minimum percentage?
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