Question

Find the critical point of the function. Then use the second derivative test to classify the nature of this point, if possible. (If an answer does not exist, enter DNE.)

f(x, y) = x² − 4xy + 2y² + 4x + 8y + 8

critical point

(x, y)=

classification ---Select--- :relative maximum, relative minimum ,saddle point, inconclusive ,no critical points

Finally, determine the relative extrema of the function. (If an answer does not exist, enter DNE.)

relative minimum value=

relative maximum value=

Answer 1

Given function is f(x,y) = x² - 4 xy + 2y² + 4x + 8y +8 Now, for (x,y) = 2x - 4y + 4 6 fg (2.7) = -42+44 + 8 __ fagy = 4 we know that for Critical points fx Gry)=0 and fy (x, y) = 0 il. 2x - 4y = - 4 - ② -42 +44 =-8 - 6 Solving ③ and 4 we get a=6 . (6,4) the critical point and I only one solution. Now, Diserăímínant of flary) at (6,4) is D (6,4) = fxx tyg - [day]² = 2.4 – 16 = -8 Lo Since D (614) 20, :. fa,y) has a saddle point. u relative minimum, relative maximum and in conclusive does not exist DNE). of we know, if D (6,4) >0 and fax (64) >0 then f has nelative minumum and if D (6, 4) >0 and for (6,4) 10 then f has relative maximum, it D (6, 4) =0 then test is inconclusive. Because