Explanation Given information: The function is f ( x , y ) = 4 x − 8 x y + 2 y + 1 . The closed triangular plate bounded by the lines x = 0 , y = 0 , and x + y = 1 . Procedure: Use the following procedure to find out the maxima and minima. Find the interior points of the region R where the function f may have local maxima and minima and evaluate f at these points. It is called critical point of f . Find the maximum and minimum value of f from the list of points that is called as absolute maximum and minimum value of the function. Calculation: Show the function f as follows: f ( x , y ) = 4 x − 8 x y + 2 y + 1 (1) Draw the bounded region in the first quadrant using the lines x = 0 , y = 0 , and x + y = 1 as shown in Figure 1. Find the interior points: Consider the function f is differentiable. Hence, f x = f y = 0 . Differentiate Equation (1) with respect to x and equate it to zero. f x = 4 − 8 y = 0 4 − 8 y = 0 y = 1 2 Differentiate Equation (1) with respect to y and equate it to zero. f y = − 8 x + 2 = 0 − 8 x + 2 = 0 x = 1 4 Find the value of function at ( 1 4 , 1 2 ) . Substitute 1 4 for x and 1 2 for y in Equation (1). f ( 1 4 , 1 2 ) = 4 ( 1 4 ) − 8 ( 1 4 ) ( 1 2 ) + 2 ( 1 2 ) + 1 = 1 − 1 + 1 + 1 = 2 Find the boundary points: Segment OA : Consider the segment OA from the Figure 1. From the segment OA , x = 0 . Show the function expressed as follows: g ( y ) = f ( 0 , y ) . Find f ( 0 , y ) using Equation 1. f ( 0 , y ) = 4 ( 0 ) − 8 ( 0 ) y + 2 y + 1 = 2 y + 1 Hence, g ( y ) = 2 y + 1 (2) For the interval 0 ≤ y ≤ 2 , g ′ ( y ) = 0 . g ′ ( y ) = 0 2 y = 0 y = 0 There is no critical point. The coordinate of the segments OA are f ( 0 , 0 ) and f ( 0 , 1 ) . Find the value of the function f ( 0 , 0 ) using Equation 1. f ( 0 , 0 ) = 4 ( 0 ) 2 − 8 ( 0 ) ( 0 ) + 2 ( 0 ) + 1 = 1 Find the value of the function f ( 0 , 1 ) using Equation 1. f ( 0 , 1 ) = 4 ( 0 ) 2 − 8 ( 0 ) ( 1 ) + 2 ( 1 ) + 1 = 2 + 1 = 3 Segment AB : Consider the segment AB from Figure 1. From the segment AB , y = − x + 1 . Show the function expressed as follows: h ( x ) = f ( x , − x + 1 )

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14.7, Problem 38E

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Determine the absolute maxima and minima of the function.

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Chapter 14.7, Problem 37E

Chapter 14.7, Problem 39E

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Chapter 14 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

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