Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \((a, b)\) is a critical point of \(f\) and both the conditions \(f_{x x}(a, b)<0\) and \(f_{y y}(a, b)<0\) hold, then \(f\) has a relative maximum at \((a, b)\).

Step-by-Step Solution

Answer

The statement is true. According to the second partial derivative test, if we have a discriminant \(D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - f_{xy}(a, b)^2\) which is positive and \(f_{xx}(a, b) < 0\), then \(f\) has a relative maximum at \((a, b)\). The given conditions \(f_{xx}(a, b)<0\) and \(f_{yy}(a, b)<0\) guarantee that their product is positive, setting the stage for a positive discriminant. Therefore \(f\) has a relative maximum at \((a, b)\).

Step 1: Understanding the Second Partial Derivative Test

The second partial derivative test is a method to determine if a critical point of a function is a relative maximum, relative minimum, or a saddle point. A critical point is a point where both the partial derivatives with respect to each variable are zero or undefined. The second partial derivative test uses the second-order partial derivatives of the function to classify the critical points. For a function \(f(x, y)\), if \((a, b)\) is a critical point, then the discriminant \(D\) is calculated as follows: \[D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - f_{xy}(a, b)^2\] If \(D > 0\) and \(f_{xx}(a, b) > 0\), then \((a, b)\) is a relative minimum. If \(D > 0\) and \(f_{xx}(a, b) < 0\), then \((a, b)\) is a relative maximum. If \(D < 0\), then \((a, b)\) is a saddle point. If \(D = 0\), the test is inconclusive.

Step 2: Analyzing the statement

We're given that both \(f_{xx}(a, b) < 0\) and \(f_{yy}(a, b) < 0\). We will now use this information to determine the discriminant \(D\). \[D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - f_{xy}(a, b)^2\] Since we don't have information about \(f_{xy}(a, b)^2\), we'll focus on \(f_{xx}(a, b)f_{yy}(a, b)\). Since both \(f_{xx}(a, b)\) and \(f_{yy}(a, b)\) are negative, their product will be positive: \[f_{xx}(a, b)f_{yy}(a, b) > 0\] Now, if the discriminant \(D(a, b) > 0\), then in addition to ourequality, it means that \(f_{xy}(a, b)^2\) is smaller than \f_{xx}(a, b)f_{yy}(a, b)\, and thus is positive. According to the second partial derivative test, we need \(f_{xx}(a, b) < 0\) for a relative maximum. We already have this condition given in the statement. So, \((a, b)\) is indeed a relative maximum.

Step 3: Conclusion

Since we found that the given conditions imply a relative maximum, the statement is true. To explain that it is true, we can say that the second partial derivative test uses the discriminant \(D\), and when the discriminant is positive and \(f_{xx}(a, b) < 0\), then the critical point \((a, b)\) is a relative maximum. In this statement, both of these conditions are met, so \(f\) has a relative maximum at \((a, b)\).

Key Concepts

Critical Point Analysis

In calculus, analyzing critical points is essential for understanding the behavior of a multivariable function. A critical point occurs where the function's first partial derivatives are zero or undefined.

This can suggest a potential peak, trough, or saddle point of the function. When you hear a function has reached a critical point, think of it as a moment of pause. It's where the function's growth or shrinkage temporarily halts, giving us clues about what happens next.

For functions of two variables, say for example a function \( f(x, y) \), a point \((a, b)\) is a critical point if both \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\).

This analysis helps us decide whether a point is a relative maximum, relative minimum, or perhaps neither, based only on the derivatives of the function.

Relative Maximum

A relative maximum of a function is a point where a function's value is higher than its neighboring points. Think of it as the top of a hill in the landscape of the function. It may not be the highest point globally, but it is higher than all the nearby points. In the context of the second partial derivative test, we identify relative maxima using both the second derivatives and the discriminant.

If at a critical point \((a, b)\), we find that both \(f_{xx}(a, b)\) and \(f_{yy}(a, b)\) are less than zero, with the discriminant \(D > 0\), we identify a local peak, or hilltop, at that point. That is, it is a relative maximum because the 'shape' of the function arcs downwards, resembling the top of a small hill.

Discriminant in Calculus

In calculus, the discriminant plays a crucial role in determining the nature of critical points for multivariable functions. The discriminant \(D\) is calculated using the formula: \[ D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - f_{xy}(a, b)^2 \]It's like a mathematical lens, allowing us to 'see' whether we face a peak, a pit, or a saddle point at a critical point.

Understanding \(D\) helps determine the type of critical point:

• If \(D > 0\) and \(f_{xx}(a, b) > 0\), it's a relative minimum.
• If \(D > 0\) and \(f_{xx}(a, b) < 0\), it's a relative maximum.
• If \(D < 0\), it's a saddle point, where the surface curves upwards in one direction and downwards in another.
• If \(D = 0\), the test is inconclusive.

Thus, the discriminant is like a decision-maker, guiding us to classify each critical point according to its surrounding surface shape.