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This activity contains 10 questions.

	The number of stationary points for f(x, y) = x3 − 6x2 + y3 − 12y2 is:   2 4 3 More than 4

	For what value of a does the pair (x, y) = (a, a + 1) maximize f(x, y) = −2x2 − 3y2 + 8x + 18y − 5?   0 −1 3 2

	For what value of a does the pair (K, L) = (a, 4a) maximize F(K, L) = 32K1/2L1/4 − 16K − 2L, K > 0, L > 0?   6 16 4 2

	The profit obtained by producing and selling x and y units respectively of two goods is given by: f(x, y) = −0.1x2 − 0.2xy − 0.2y2 + 47x + 48y − 600. Find the profit maximizing quantities.   (x, y) = (230, 5) (x, y) = (195, 5) (x, y) = (5, 230) None of the above

	Which of the following statements is incorrect?   A function f(x) defined on a closed bounded set S can only have a maximum point in S if f(x) is continuous in S If f(x) has a maximum at a, then g(x) = −f(x) has a minimum at a A differentiable function f(x) can only have a maximum point at an interior point a of its domain if a is a stationary point Given a function f(x) on a convex domain S, a sufficient condition for it to attain a maximum at an interior stationary point a is that f is concave

	Which of the following sets fails to be both closed and bounded?   { (x, y) : 4 < x2 + y2 < 9 } { (x, y) : 2x + 3y < 6, x > 0, y > 0 } { (x, y) : x3 + y3 < 2 } { (x, y) : x2 + y2 < 9 }

	If f(x, y) > 0 for all (x, y) and f has a maximum at (a, b), which of the following functions does not necessarily have a maximum at (a, b)?   (f(x, y))3 (f(x, y))−1 ln f(x, y) ef(x, y)+2

	The stationary points of f(x, y)are of the following type:   One local maximum point and one local minimum point One local maximum point and one saddle point Two saddle points Two local maximum points

	A firm uses capital K and labour L to produce units of a commodity. The price for each unit produced is p, the price of capital is r and the price of labour is w. The profit is then All variables and all constants are positive. Denote the optimal values by K*, L*, and Q* and let the associated optimal profit be *. Which of the following equalities is false?   (A) (B) (C) (D)

	The function f(x, y) = 3x2 − 3xy2 + y3 + 3y2 has three stationary points: (0, 0), (2, 2), and (1/2, −1). The maximum value of f(x, y) defined on the square S = [0, 3] × [0, 3] occurs at:   (2, 2) a point on the edge x = 0, y ∈ [0, 3] a point on the edge x = 3, y ∈ [0, 3] a point on the edge x ∈ [0, 3], y = 0

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