A function f: R2 → R is said to be independent of the second variable if for each x € R we have f (x, y1) = f (x, y2) for all y1, y2. €R Show that f is independent of the second variable if and only if there is a function f: R→R such that f(x, y) = g(x). What is f1 (a, b) in terms of g1?
Answer to relevant QuestionsDefine when a function f: Rn → R is independent of the first variable and find f1 (a, b) for such f. Which functions are independent of the first variable and also of the second variable?Let f: R→R 2. Prove that f is differentiable at a € R if and only if f 1 and f 2 are, and in this case f 1(a) = ((f 1)1 (a) (f 2)1 (a)).Regard an n x n matrix as a point in the -fold product Rn x . x Rn by considering each row as a member of Rn.. a. Prove that det : Rn x . x Rn → Rn is differentiable and b. If aij : R →R are differentiable ...Find the partial derivatives of f in terms of the derivatives of g and h ifA function f: Rn → R is is homogeneous of degree m if f (tx) = tmf(x) for all x and t. If f is also differentiable, show that
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