Explanation Given information: The function is f ( x , y ) = x 2 + x y + y 2 + 3 x − 3 y + 4 Critical point: An interior point of the domain of a function f ( x , y ) where both f x and f y are zero or where one or both or where one or both of f x and f y do not exist is a critical point of f . Theorem:11 Suppose that f ( x , y ) and its first and second partial derivative is continuous throughout a disk centered at ( a , b ) and that f x ( a , b ) = f y ( a , b ) = 0 . i. f has a local maximum at ( a , b ) if f x x < 0 and f x x f y y − f x y 2 > 0 at ( a , b ) . ii. f has a local minimum at ( a , b ) if f x x > 0 and f x x f y y − f x y 2 > 0 at ( a , b ) . iii. f has a saddle point at ( a , b ) if f x x f y y − f x y 2 < 0 at ( a , b ) . Calculation: Provide the function as shown below: f ( x , y ) = x 2 + x y + y 2 + 3 x − 3 y + 4 (1) Differentiate Equation (1) with respect to x . ∂ f ∂ x = ∂ ∂ x ( x 2 + x y + y 2 + 3 x − 3 y + 4 ) = 2 x + y + 3 (2) Differentiate Equation (1) with respect to y. ∂ f ∂ y = ∂ ∂ y ( x 2 + x y + y 2 + 3 x − 3 y + 4 ) = x + 2 y − 3 (3) Find the critical point as shown below: Equate the Equation (2) and (3). 2 x + y + 3 = x + 2 y − 3 = 0 2 x − x + 3 = 2 y − y − 3 = 0 2 x − x + 3 = 0 x + 3 = 0 x = − 3 2 y − y − 3 = 0 y − 3 = 0 y = 3 Thus, the critical points is ( − 3 , 3 )

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 1E

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Calculate the local maxima, minima, and saddle point of the function.

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Chapter 14.6, Problem 61E

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A cook has finished baking a cake and placed it on the bench to cool. The temperature in the room is 20°C and the temperature of the cake when it was taken out of the oven is 160°C (a) Given that the temperature of the cake is governed by Newton's law of cooling, write down a differential equation governing T(t), the temperature of the cake after t hours. What is the appropriate initial condition? (Newton's law of cooling: dT dt =-K(T-Ta), where K is a constant and Ta is the ambient temperature.) (b) From you answer in part (a), derive the solution T(t) = 20 + 140e Kt, where K is a (c) constant. Given that the cake has cooled to 90°C after 1 hour, determine the constant K. (d) The cook decides that the cake is cool enough to be taken out of the cake pan when its temperature lowers to 40 degrees C. Find when this will happen, both in exact form and as a decimal approximation to at least 2 decimal places, showing all working.

Chapter 14 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

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Calculate the local maxima, minima , and saddle point of the function.

Calculate the local maxima, minima, and saddle point of the function.

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Chapter 14 Solutions