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. • The cross-section models are 2-D functions, and 2-D functions have local extrema where their derivative = 0. • The point (a, b) is a critical point of z = f ( x, y) if fx ( a, b) = 0 and fy ( a, b) = 0 . • 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 • é f xx First we find the Hessian: ê ê f yx ë f xy ù ú f yy ú û • 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? 2 2