Question

Sheet A company produces units of product and units of product (both in hundreds per month). The monthly profit equation (in thousands of dollars) is given by the following: P(x,y) - 4x2 + 4xy - 3y2 + 4x + 10y + 81 Find and evaluate . Use formulas to get an exact answer! Pxx.y) Px(1,3) What does this value mean? Answer in this textbox. Answer: What is the critical point of? What are the equations for, and ? o po DO Critical point: PxyiX. Pxxix.) Pyyix.) How many of each product should be produced each month to maximize profit? What is the maximum profit? Maximum profit

Answer 1

solution. - nation is Morthely profit Equ platy) = -4x² + 4xy - 3y + 40+10y + 8 Number of products of type ady de de are Now is da we can solve this problem by using: Maximum and minimum of Play) that multivariable function, How to find Maximum Menimum value of Multivariable function Steps Involved are : 0 Find op ap sy. Equate both equal - al to Zero to get sy. & op ay ♡ Find D: 3 3 Top 2x² 242 Lagay 6 so, If & play) 70 gep Play) ~ Max CS x & 22 0 Scanned with CamScanner 222

IF, Dla,y) >0 & 3p yo Play) 2x2 - Minimum Conditions will critical These both potonts (any). give And In Last I Dla, y) <o saddle points, Here. Por 2x2 [x,y) 22p Pelny sse op P (244) g² p ay Pyy (1,9) ay² representing partial derivatives] . d 2x So Now Follow these steps. For our original Function profit aequation. Plag) = - 44 ² + 4014 - 3y² + 4x+10y +81 Note: Differentiation of constants are Scanned with CamScanpro. always CS

So Of Pe (2,4)= [ differentiation Ja ap Play) , keeping y - Constant [so, differentiation of terms of Yonly) will be o > f = hela, y -442*x + 4 4 + +4 Po(a,y) -8x + 4y + 4 © So, at 4 § (113) py (1,3) -811 + 413) +4. 16-8 = ㅋ px 0,3) Means Test Play is the rate of change profit Equation with respect to e-sonly are only considering change and keeping y Constant How profits Actually depends only on we It will tell us H. CS Scanned with CamScanner

wow Py laty) differentiation of play) keeping xa constant. Pula,y) = op ay 0 + 4% - 6y +10. Poy (a,y) = 44- by tot 10 (1) So, o e 9 Now to find critical points. 2P OP O2 sy P (4) P (ny) -8x + 4y + 450 4x-by + 10 =0. Solve these equations to get (19) sulteply second Equation by 21401- by tio) + 8x + 4y +4 8x8x - 124 +44 + 2014 Cs Scanned with 84 +24=0 -> y = 3 mu 2 and Add = 0.

put y = g in Frost Equatim, - 8xxt 4/3) + 4 = 0 - 8x + 12+ 4 =0 -8x = -16 a=2 so point is + Dogs (aig (2,3) Noro Find D - step - 4 Scritical - D = 9² P 22 p point? D 272 Biz Laxay So, find : Pax Pyy Pay ap ox Po ax Pare (94) = 3[F-8&thy te a Paraply) = -6 Ja Pay (ory) 2 [2x=by to да ay dat Pay (1.4) Pay (riy) = 4 *. 4 CS Scanned with CamScanner

Note 8 Pay (9,4) Pyxlay) or 2²p 22p sady ayon. dy 24a-by to gy sy 1 Pgy (ay) put Au these values in. Don = D = Pad Pyy (8) 1-6) [45? 48-16 D = 32 70 (213) is So, Local Maxiona, a P (2,3) to iu give maximum value. Number of each each products, 2 both are 3 in hundred pro fit P L213) = -499 +46][3) -3483 + 4 L21101318 -414) +24-27+ 8+30+81 CS Scanned with Cants artlanum profito thousands of dollar y Maximum 81

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