Question

please solve all parts.

For the function f(x,y)=x3+y3 – 6y2-3x+5, do the following: (a) Determine its critical point(s) if exists. Express your answer as coordinate pairs with parentheses and commas. Separate your answers with commas and list in ascending order of x if the function has more than one critical point. Use **DNE" if the function has no critical point. Answer: (b) Use the D-Test to classify at each critical point whether the function has a relative maximum or minimum, or a saddle point, or inconclusive: At the first critical point, the D value is ---Select-- y, and the second order partial derivative xx is --Select- Therefore at this point ---Select-- , and the second order partial derivative fxx is ---Select-- . At the second critical point, the D value is ---Select Therefore at this point ---Select--- , and the second order partial derivative xx is --Select- At the third critical point, the D value is ---Select Therefore at this point ---Select--- , and the second order partial derivative fox is --Select.. . At the fourth critical point, the D value is --Select Therefore at this point ---Select---

Answer 1

flory) = apy ? by²onta To And the Citical points fr= 3x2-3 fy = 3y by fax= 6a, fyy = 64-12, fay = 0 Let us first find critical points. Critical points will be solution to bystem of cgn. fr=392-3.30 ty= 3y = 1 ay so. 30 = 3 34²1 ay so n², 3y = 4y] . _ 192 £J 1 39 [y-43 to ly=0 yeh Here we get the critical fornts as 4, combinations (1,0) (1,4) G1,0), (1,4) 3) Now by D test we can find Relative Maximum of minimum or baddle pont At (110), the A value is P= fxx fyy - (fxy] 2 1 = (6) (64-12) - [012 A = 64 363y = 4202) At Cipol, D= 36(0) - 72 (1) D = -722 <o. So at (,0) Dis Negative, fxx is the CS Scanned with So this is Saddle point. Catheefunction has saddle point at (1,0).

At 6,4) D = 3624 -728 At (,,H), De 36(1)(3) ~ 72(1) = 144-72 D 2722]>o. So at (114) D is positive, fxx s. 6x = 6(1) >o the so this to fonction has relative minimum at (114) At Girol D = fxx fyy (fxy) " D 36xy – 72 D=86(-1) (0)-7261) P = 72 %0. - fxx is for = 6(-1)=-620. to this fonction has thous that no conclusion by 8 test I I AT (1,4) D = fxx fiyy - (fxy) 2 = 36 xy – 724 = 36(1)G) -7267). = +44 +72 De-72 20. fxx is 6x = 6 (-1)=-62o. to this fonction has a relative Maximum at (14) Scanned with CamScanner