(a) Consider the function f (x, y) = ax2 + 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) = ax2 + 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) = a1x12+ a2x22+…+an xn2 , 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.
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