(a) Consider the function f (x, y) = ax 2 + by2, where a and b are nonzero constants

Chapter 4, Problem 23

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(a) Consider the function f (x, y) = ax 2 + by2, where a and b are nonzero constants. Show that the origin is the only critical point of f , and determine the nature of that critical point in terms of a and b. (b) Now consider the function f (x, y,z) = ax 2 + by2 + cz2, where a, b, and c are all nonzero. Show that the origin in R3 is the only critical point of f , and determine the nature of that critical point in terms of a, b, and c. (c) Finally, let f (x1, x2,..., xn) = a1x 2 1 + a2x 2 2 ++ an x 2 n , where ai is a nonzero constant for i = 1, 2,..., n. Show that the origin in Rn is the only critical point of f , and determine its nature.