Find the critical points of the following functions. Use the Second Derivative Test to determine whether each critical point corresponds to a loal maximum, local minimum, or saddle point. f(x,y)=x^4+2y^2-4xy

a2linetagadaW 2020-11-26 Answered
Find the critical points of the following functions. Use the Second Derivative Test to determine whether each critical point corresponds to a loal maximum, local minimum, or saddle point.
f(x,y)=x4+2y24xy
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Expert Answer

lamusesamuset
Answered 2020-11-27 Author has 93 answers

Let c=(a,b) point of the given function. Then we have
(x4+2x24xy)xa,b=0 and (x4+2x24xy)ya,b=0
which further implies that (4x34y)a,b=0 and (4y4x)a,b=0 implying that a3b=0 and b-a=0 leading to (a,b)(0,0),(1,1),(1,1).
Therefore the required critical points for the given function are (0, 0), (1, 1), and (-1, -1).
Now we calculate the values of 2fx2,2fy2 and 2fxy to be able to use the second derivative test. We get 2fx2=x(4x34y),2fy2=y(4y4x), and 2fxy=x mplying that 2fx2=12x2,2fy2=4, and 2fxy=4.
Now, we denote D(c)=(12x2c)(4c)((4)c)2
Now we calculate this for all the above three critical points we found as
D(0,0)=(12(0)24(4)2=16).
D(1,1)=(12(1)24(4)2=31).
D(1,1)=(12(1)24(4)2=31).
We also have 2fx2

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