Question

Question 17 Find and classify the critical point of f (, y) = 263 – 25y? – 12xy - 11x² + 46y +389. To find the critical point, you must solve two equations. Type them below: Solve this system and round your answers to the hundredths place: Find the third coordinate of the critical point: Calculate the determinant of the second-order partial derivative matrix Fill in each of the remaining spaces with a word or phrase from the following list: • positive • negative • zero • maximum • minimum • saddle point • not needed The critical point is a because Dis and faxis

Answer 1

flry) = 26x— 2542-1284-114*+64 +389 - Apply partial derivative to the given Equation and set it Equal to zero fr=0 s (262–2542–12x4 - 1182 +H5y 389) =0 of 26111 – 0 -121114 – 11(22) toto -0 . Y terms are constant 26-128 -22% =0 L = noun L a constant to U fy 50 fot ( 261 – 259*-1214 -1122+464 +389) = 0 S 0-25(24) – 12x61) - 0 + 46.17 to 50 f - 504-122+66 =0 Two equations -222-124 +26 = 0 -12% -50y +46 =0 1-114-64+13 = 01- 10 -62—254 +2350) 7° (OR) Solve equction © and o we geb 0x6 =) -664-364 + 78 50 @x11 -662-2754+253 = 0 2394–175 = 0 y = 175 = 0.7322175 239 05 -13-60170) +l3 so 239 -111- 1050 +13 = 0 - 2057 239 -13 + 1050 Lila - - 3107 +1050 - -112 - -13+ -15239 - = 239 239 239 11a = 2057 87= 2057 - 2054 = 0.782426 x = 70-78) y = 10.73 239X1) 2629 f(0-78, 0-73) = f = 2610-78) -2560-73)-1210-78)(0.73) – 11(0.78 +4610-73) +389 f = 20-28 - 13-3225-8668328 – 6+6924+33.58 +389 P = 416.1098) (OR) (F = H16-11

frax = of da (fx) = ( 26–124–221) fan = 0-0–2211) 5-22 Pyy = (fy) = are in ( -50y =12x +46) = -500–0+0. fyy = -50 fry role in (hu) = f ( 26-12% -294) = 0 21201-O =-12 fyx = som del fy) = o p (= 50y =12x+46) = 0 -12013 to fya = -12 Now i operator is given by D = (for) (f44) – [fxy)? 0 Com conditions from and derivative test . If oko I critical points reffers to Saddle Points fan >o , critical points reffers to Loed minimum p fyy ko, Critical points reffers to local maximum Critical point (Mg) = (0.78, 0.73) fax = -22, fyy=-50, fry = -12 D = 622)(-50) – Klej= 1100-144 =1956)>0 - D=1956 30, fyy ko Hence Gritical point (0.78, 0.73) reffers to local maximum. The Critical point is a local manimum because D is positive and fun is negative and fyy is negative