Find the two critical points of the following function, compute the deter- minan

Question

Find the two critical points of the following function, compute the deter- minant of the Hessian matrix H at each point and classify each point as a maximum, a minimum or a saddle point. Enter your answers starting with the stationary point with the smallest z value. Smaller z-value det(H) = Classify the critical point Maximum Minimum saddle point Larger r-value det(H)- Clasify the critical point Maximum Minimum saddle point Submit Answer Save Progress Question 1 of 8 View Next Question a Type here to search

Explanation / Answer

f(x,y) = 3x²+3xy²+y^3

find partial derivative as

fx= 6x+3y²

fy = 6xy +3y²

To find critical point solve fx =0 and fy= 0

6xy +3y² =0

6x+3y² = 0

Subtrcat both

6xy -6x =0

6x(y-1) =0

x =0 or y=1

Hence critical points are (0,0) and (-1/2, 1)

Find detrminant of hesian matrix as

det(H) = 6*(6x+6y) -(6y)(6y) = 36(x+y-y²)

Smaller x-value :

x = -1/2

y = 1

Det(H) = -18

Saddle point

---------

larger x-value :

x = 0

y = 0

Det(H) = 0

Saddle point

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