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Exercise 5. (From Ira Rosenholz in the Problems Section of the Mathematics Magazine, Vol. 60 NO. 1, February 1987.) The book Calculus in Vector Spaces by Corwin and Szczarba contains the following in its discussion of extrema for functions of several variables: "Suppose f has local marima at vi and va. Then f must have another point, v3, because it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between mountains) or a local minimum (a true valley)." (a) Using the function f(z, y) = 4r?e" - 2r - etv, show that the impossible is possible (that is, there exists a function with two local maximum points and no other critical points). (b) Use appropriate technology, (for example, geogebra) to plot this function. Is there a "valley" between the two maximum points? Where did the authors of the calculus book go wrong in their argument?

Exercise 5. (From Ira Rosenholz in the Problems Section of the Mathematics Magazine, Vol. 60 NO. 1, February 1987.) The book Calculus in Vector Spaces by Corwin and Szczarba contains the following in its discussion of extrema for functions of several variables: "Suppose f has local marima at vi and va. Then f must have another point, v3, because it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between mountains) or a local minimum (a true valley)." (a) Using the function f(z, y) = 4r?e" - 2r - etv, show that the impossible is possible (that is, there exists a function with two local maximum points and no other critical points). (b) Use appropriate technology, (for example, geogebra) to plot this function. Is there a "valley" between the two maximum points? Where did the authors of the calculus book go wrong in their argument?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 15EQ
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Exercise 5. (From Ira Rosenholz in the Problems Section of the Mathematics Magazine, Vol. 60 NO. 1, February 1987.) The book Calculus in Vector Spaces by Corwin and Szczarba contains the following in its discussion of extrema for functions of several variables: "Suppose f has local marima at vi and v2. Then f must have another point, v3, because it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between mountains) or a local minimum (a true valley)." (a) Using the function f(r, y) = 4r?e – 2r - ety, show that the impossible is possible (that is, there exists a function with two local maximum points and no other critical points). (b) Use appropriate technology. (for example, geogebra) to plot this function. Is there a "valley" between the two maximum points? Where did the authors of the calculus book go wrong in their argument?
Transcribed Image Text:Exercise 5. (From Ira Rosenholz in the Problems Section of the Mathematics Magazine, Vol. 60 NO. 1, February 1987.) The book Calculus in Vector Spaces by Corwin and Szczarba contains the following in its discussion of extrema for functions of several variables: "Suppose f has local marima at vi and v2. Then f must have another point, v3, because it is impossible to have two mountains without some sort of valley in between. The other critical point can be a saddle point (a pass between mountains) or a local minimum (a true valley)." (a) Using the function f(r, y) = 4r?e – 2r - ety, show that the impossible is possible (that is, there exists a function with two local maximum points and no other critical points). (b) Use appropriate technology. (for example, geogebra) to plot this function. Is there a "valley" between the two maximum points? Where did the authors of the calculus book go wrong in their argument?
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