2 y x x x f x x f 296 166 2 2 2 y x y y f y y f 29

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Finding Relative Maxima & MinimaEXAMPLEEXAMPLEA monopolist manufactures and sells two competing products, call them I and II, that cost $30 and $20 per unit, respectively, to produce. The revenue from marketing xunits of product I and yunits of product II is Find the values of xand ythat maximize the monopolist’s profits. (29.2.01.004.011298,22yxxyyxyxR---+=
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 34 of 62Finding Relative Maxima & MinimaNow we set both partial derivatives equal to 0 and then solve each for y.CONTINUEDCONTINUEDNow we may set the equations equal to each other and solve for x.02.004.098=--xy04.004.0112=--yx24505+-=xy2801.0+-=xy2801.024505+-=+-xx28024509.4=+-x21709.4-=-x443x
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 35 of 62Finding Relative Maxima & MinimaWe now determine the corresponding value of yby replacing xwith 443 in the equation y= -0.1x+ 280.CONTINUEDCONTINUED(292362804431.0+-=ySo we now know that revenue is maximized at the point (443, 236). Let’s verify this using the second-derivative test. To do so, we must first calculate (29.,222222-=yxRyRxRyxD
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 37 of 62§7.4Lagrange Multipliers and Constrained Optimization
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 38 of 62Background and Steps for Lagrange MultipliersUsing Lagrange MultipliersLagrange Multipliers in ApplicationSection Outline
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 40 of 62Steps For Lagrange MultipliersL-1L-2L-3
Using Lagrange MultipliersEXAMPLEEXAMPLESOLUTIONSOLUTIONMaximize the function , subject to the constraint22yx+

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