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Chapter 11: Problem 36

State the Second Partials Test for relative extrema and saddle points.

Short Answer

Expert verified
The Second Partials Test for relative extrema and saddle points states that if the determinant of the Hessian Matrix \( D = f_{xx}(x, y)f_{yy}(x, y) - f_{xy}^2(x, y) \) is greater than 0, and \( f_{xx} > 0 \), then the function has a local minimum. If \( D > 0 \) and \( f_{xx} < 0 \), it has a local maximum, and if \( D < 0 \) it has a saddle point.

Step by step solution

01

Defining Relative Extrema

Relative extrema are the 'high' and 'low' points of a curve. Formally, a function \( f(x, y) \) has a relative maximum at a point \( (x, y) \) if \( f(x, y) \) is greater than or equal to the value of \( f \) at all points in the vicinity of \( (x, y) \). Similarly, \( f(x, y) \) has a relative minimum if \( f(x, y) \) is less than or equal to the value of \( f \) at all points in the vicinity of \( (x, y) \).
02

Defining Saddle Points

A saddle point of a function is a point in the domain where the function neither has a relative maximum nor minimum, but the directional derivatives of the function changes signs. Hence, in some directions it acts like a relative maximum and others as relative minimum.
03

Second Partials Test

The second partials test uses the second partial derivatives of the function to determine the nature of the critical points. Suppose \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \) are the second derivative of the function \( f \) and \( D = f_{xx}(x, y)f_{yy}(x, y) - f_{xy}^2(x, y) \) is the determinant of the Hessian Matrix. If \( D > 0 \) and \( f_{xx}(x, y) > 0 \), then \( f \) has a local minimum at \( (x, y) \). If \( D > 0 \) and \( f_{xx}(x, y) < 0 \), then \( f \) has a local maximum at \( (x, y) \). If \( D < 0 \), then \( f \) has a saddle point at \( (x, y) \).

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