Question

In Exercises \(45-50,\) find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. $$ f(x, y)=x^{3}+y^{3} $$

Short answer

The function has a single critical point at (0, 0) for which the Second Partials Test fails to determine the nature of the point.

Step-by-step solution

Calculate the first derivatives

Find the partial derivatives with respect to x and y. The first derivative with respect to x is \(f_{x}(x, y) = 3x^{2}\). Similarly, the first derivative with respect to y is \(f_{y}(x, y) = 3y^{2}\).

Find the critical points

At critical points, the derivatives are zero. So, set \(f_{x}(x, y) = 0\) and \(f_{y}(x, y) = 0\) to get the points (x, y) where both are true. On solving, we find that the only critical point is (0, 0).

Conduct the Second Partials Test

Now calculate the second partial derivatives: \(f_{xx}(x,y) = 6x, f_{yy}(x, y) = 6y\) and \(f_{xy}(x, y) = f_{yx}(x, y) = 0\). Now, for the Second Partials Test, we find the determinant of the second derivative matrix, \(D = f_{xx}(x, y) * f_{yy}(x, y) - (f_{xy}(x, y))^2 = 0\), at the critical point (0, 0).

Evaluate the test result

Since the determinant at the critical point (0, 0) is zero, the Second Partials Test fails to determine whether this point is a local extremum or not.

Key concepts

First Partial Derivatives

Understanding the concept of first partial derivatives is critical in multivariable calculus, especially when looking for critical points of a function with multiple variables. Think of the process as slicing the function into one-variable functions, and then analyzing them independently.

For a two-variable function like \( f(x, y) \), we compute two first partial derivatives: \( f_x \) with respect to x and \( f_y \) with respect to y. These derivatives represent the instantaneous rate of change of the function as one variable changes while the other is held constant. In mathematical terms:

• \( f_x(x, y) = \frac{\text{\text{d}}}{\text{\text{d}}x} f(x, y) \), the derivative of the function with respect to x.
• \( f_y(x, y) = \frac{\text{\text{d}}}{\text{\text{d}}y} f(x, y) \), the derivative of the function with respect to y.

Setting both \( f_x \) and \( f_y \) to zero leads us to critical points. These are the points where the function may achieve local maxima, minima, or saddle points. Essentially, this is where the 'tangent' plane to the surface of the function is horizontal.

Second Partials Test

The Second Partials Test is the multivariable counterpart to the second derivative test from single-variable calculus. It's used to classify critical points — the potential maxima, minima, or saddle points.

To perform this test, we calculate the second-order partial derivatives: \( f_{xx} \), \( f_{yy} \), and the mixed partials \( f_{xy} = f_{yx} \). With these, we find the determinant of the second derivative matrix (also known as the Hessian) at the critical point:
\[ D = f_{xx}(x, y) * f_{yy}(x, y) - (f_{xy}(x, y))^2 \]
If \( D > 0 \) and \( f_{xx} > 0 \), we have a local minimum; if \( D > 0 \) and \( f_{xx} < 0 \), a local maximum is found; and \( D < 0 \) indicates a saddle point. However, if \( D = 0 \), the test is inconclusive — this means more investigation is needed to understand the nature of the critical point.

Relative Extrema

Relative extrema (or local extrema) are points where a function reaches local highs (maxima) or lows (minima) within a neighborhood around that point. In the territory of functions with several variables, finding these points involves a combination of using partial derivatives and tests like the Second Partials Test.

Upon identifying critical points by setting the first partial derivatives to zero, one must assess these points to conclude whether they constitute relative maxima, relative minima, or neither (which could possibly be saddle points or points of inflection).

Remember that the existence of relative extrema is deeply connected to the behavior of the function in the region close to the critical points. By analyzing the changes in the directional derivatives nearby, we can gauge how the function evolves - does it go up, go down, or twist away? It's this local behavior that defines the 'relativity' of the extrema.