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+2y2−4xy

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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+2y2−4xy

lamusesamuset · Accepted Answer

Let c=−(a,b) point of the given function. Then we have ∂(x4+2x2−4xy)∂x∣a,b=0 and ∂(x4+2x2−4xy)∂y∣a,b=0 which further implies that (4x3−4y)∣a,b=0 and (4y−4x)∣a,b=0 implying that a3−b=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 ∂2f∂x2,∂2f∂y2 and ∂2f∂x∂y to be able to use the second derivative test. We get ∂2f∂x2=∂∂x(4x3−4y),∂2f∂y2=∂∂y(4y−4x), and ∂2f∂x∂y=∂∂x mplying that ∂2f∂x2=12x2,∂2f∂y2=4, and ∂2f∂x∂y=−4. Now, we denote D(c)=(12x2∣c)(4∣c)−((−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 ∂2f∂x2∣

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

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