Find the three critical points of the following function, compute the de- termin

Question

Find the three critical points of the following function, compute the de- terminant of the Hessian matrix H at each point and classify each point as a maximum, a minimim or a saddle point. Enter your answers starting with the stationary point with the smallest z value. Rx, y) = 2x. + 4x, + y. Smallest z-value det(H) = Classify the critical point Maximum Minimum saddle point Intermediate r-value det(H) = Classify the critical point Maximum Minimum saddle point Largest a-value det(H)= Classify the critical point Maximum Minimum saddle point

Explanation / Answer

f(x,y) = 2x^4+4x²y+y^4

find partial derivatve as

fx= 8x^3 +8xy = 8x(x² +y)

fy = 4x² +4y^3 = 4(x²+y^3)

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

8x(x² +y) = 0

4(x²+y^3)=0

Solving both we obatin solutions are

(-1, -1), (0,0) and (1,-1)

Determinant of Hessian matrix is 288x²y² -64x² +96y^3

Now

Smallest x -value

x= -1

y= - 1

det(H) = 128

MINIMUM

-------------

Intermediate x -value

x= 0

y= 0

det(H) = 0

Saddle point

--------------

Largest x -value

x= 1

y= - 1

det(H) = 128

MINIMUM

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