A function is given by,

f(x,y) = x^4 - y^2 - 2x^2 + 2y - 7

Using the second derivative test for functions of two variables, classify the points (0,1) and (-1,1) as local maximum, local minimum or inconclusive.

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(-1,1) is classified as a saddle point because the value it gives after the second derivative test is less than 0 therefore the value is inconclusive. While (0,1) is classified as a local maximum because the value it gives after the second derivative test is a negative.

To classify the points (0,1) and (-1,1) using the second derivative test, we need to calculate the second-order partial derivatives and evaluate them at these points.

Step 1: Calculate the first-order partial derivatives:
fx = d/dx (x^4 - y^2 - 2x^2 + 2y - 7) = 4x^3 - 4x
fy = d/dy (x^4 - y^2 - 2x^2 + 2y - 7) = -2y + 2

Step 2: Calculate the second-order partial derivatives:
fxx = d/dx (4x^3 - 4x) = 12x^2 - 4
fyy = d/dy (-2y + 2) = -2
fxy = d^2/dxdy (x^4 - y^2 - 2x^2 + 2y - 7) = 0 (since the derivatives of fx and fy with respect to the other variable cancel each other out)

Step 3: Evaluate the second-order partial derivatives at the given points:
For the point (0,1):
fxx(0,1) = 12(0)^2 - 4 = -4
fyy(0,1) = -2
For the point (-1,1):
fxx(-1,1) = 12(-1)^2 - 4 = 8
fyy(-1,1) = -2

Step 4: Apply the second derivative test:
For a point to be a local maximum, fxx < 0 and fxx*fyy - (fxy)^2 > 0.
For a point to be a local minimum, fxx > 0 and fxx*fyy - (fxy)^2 > 0.

For the point (0,1):
fxx(0,1) = -4 (negative)
fxx(0,1)*fyy(0,1) - (fxy(0,1))^2 = (-4)(-2) - (0)^2 = 8 (positive)
Since fxx is negative and fxx*fyy - (fxy)^2 is positive, the point (0,1) is classified as a local maximum.

For the point (-1,1):
fxx(-1,1) = 8 (positive)
fxx(-1,1)*fyy(-1,1) - (fxy(-1,1))^2 = (8)(-2) - (0)^2 = -16 (negative)
Since fxx is positive and fxx*fyy - (fxy)^2 is negative, the point (-1,1) is classified as inconclusive.

Therefore, the point (0,1) is a local maximum, and the point (-1,1) is inconclusive.