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Sec 15.7 Maximum and Minimum Values

Sec 15.7 Maximum and Minimum Values. Definition: A function of two variables has a local maximum at ( a , b ) if f ( x , y ) ≤ f ( a , b ) when ( x , y ) is near ( a , b ). The number f ( a , b ) is called a local maximum value .

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Sec 15.7 Maximum and Minimum Values

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  1. Sec 15.7 Maximum and Minimum Values Definition: A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). The number f (a, b) is called a local maximum value. A function of two variables has a local minimum at (a, b) if f (x, y) ≥ f (a, b) when (x, y) is near (a, b). The number f (a, b) is called a local minimum value. 3. If the inequalities in 1. and 2. hold for all points in the domain of f , then f has an absolute maximum (or absolute minimum) at (a, b).

  2. Theorem: If f has a local maximum or minimum at (a, b) and the first-order partial derivatives of f exist, then i.e. Definition: A point (a, b) is called a critical point (or a stationary point) of f if or if one of these partial derivatives does not exist.

  3. Second Derivative Test Suppose the second partial derivatives of f are continuous on a disk with center (a, b), and suppose [i.e (a, b) is a critical point]. Definition: In case c), the point (a, b) is called a saddle point of f. Note: If D = 0, the test gives no information.

  4. Extreme Value Theorem for Functions of Two Variables If f is a continuous on a closed, bounded set D in , then f attains an absolute maximum value and an absolute minimum value at some points in D. Finding the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: Find the values of f at the critical points of f in D. Find the extreme values of f on the boundary of D. 3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

  5. Sec 15.8 Lagrange Multipliers Method of Lagrange Multipliers: To find the maximum and minimum values of f (x, y, z) subject to the constraint g(x, y, z) = k [ assuming that these extreme values exist and on the surface g (x, y, z) = k ]: (a) Find all the values of x, y, z, and λ such that and g(x, y, z) = k Evaluate f at all the points (x, y, z) that results from step (a). The largest of these values is the maximum value of f ; the smallest is the minimum value of f .

  6. Two Constraints To find the maximum and minimum values of f (x, y, z) subject to two constraints g(x, y, z) = k and h(x, y, z) = c. (a) Find all the values of x, y, z, λ , and μ such that g(x, y, z) = k and h(x, y, z) = c . Evaluate f at all the points (x, y, z) that results from step (a). The largest of these values is the maximum value of f ; the smallest is the minimum value of f .

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