Answer
See explanation
Work Step by Step
Step 1: Second Partial Derivative Test Recall the second partials test. Let \((x_0, y_0)\) be a critical point of a function \(f(x, y)\), and \[D = f_{xx}(x_0, y_0)f_{yy}(x_0, y_0) - f^2_{xy}(x_0, y_0)\] If \(D > 0\) and \(f_{xx}(x_0, y_0) > 0\) or \(D > 0\) and \(f_{yy}(x_0, y_0) > 0\), then \((x_0, y_0)\) is a relative minimum. If \(D > 0\) and \(f_{xx}(x_0, y_0) < 0\) or \(D > 0\) and \(f_{yy}(x_0, y_0) < 0\), then \((x_0, y_0)\) is a relative maximum. If \(D < 0\), then \((x_0, y_0)\) is a saddle point, and if \(D = 0\), the test is inconclusive. Step 2: If the Second Partials Test is Inconclusive If the second partials test is inconclusive, we have to resort to other methods. We can take a look at the contour plot or 3D graph if technology is available, or we could introduce some algebraic tricks or manipulation. Step 3: Investigating \(D = 0\) If \(D = 0\) at the point \((x_0, y_0)\), then we can investigate the values of the function at \((x_0 + a, y_0 + b)\). We can then try to investigate the behavior of the function for arbitrary values of \(a\) and \(b\), inspired by the first derivative test in single-variable calculus. Step 4: Determining Relative Extrema for \(D = 0\) If the function appears to decrease for every choice of \(a\) and \(b\), then \((x_0, y_0)\) is a relative maximum. If it increases, then \((x_0, y_0)\) is a relative minimum. If we select values of \(a\) and \(b\) such that the function appears to decrease in one direction and increase in another, then \((x_0, y_0)\) is a saddle point.
