Review exercises
Work Sheet 1
Monday 24 Jan. 2016
1. (Midterm 1, Oct. 2014)Assume that f (x, y) has continuous partial derivatives of
all orders, and
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fyx (x, y) = exy .
Compute fxyy . State in detail any result that you use.
2. (Sample Midterm 5) Given functions:
F (x, y) = x + ey , G(x, y) = y + ex .
Does there exist a function f (x, y) such that ∇f (x, y) = hF, Gi? Justify your
answer and clearly state any result that you may use.
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Review exercises
Work Sheet 1
Monday 24 Jan. 2016
3. Suppose f (x, y) = x2 − 2x + y 2 + 2y + 5.
a) Find all critical points of f and classify each point as a local minimum, local
maximum, or saddle point.
b) Find the absolute maximum and minimum values of f on the region R where
R is the closed region bounded by the triangle with vertices (−1, 0), (3, 0) and
(3, 1).
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Review exercises
Work Sheet 1
Monday 24 Jan. 2016
c) (Example 7 page 946 of the textbook)
Find the absolute
maximum and minimum values of f on the region S = (x, y) : x2 + y 2 ≤ 4 .
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Review exercises
Work Sheet 1
Monday 24 Jan. 2016
4. (Midterm 1, Oct 2014) A company produces:
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P (x, y) = 5x 5 y 5 .
units of goods per week, utilizing x units of labour and y units of capital. If labour
costs 1 dollar per unit, and capital costs 1 dollar per 8 units, use the method of
Lagrange multiplier to find the most cost-efficient division of labour and capital
that the company can adopt if its goal is to produce 80 units of goods per week.
Clearly state the objective function and the constraint. You are not required to
justify that the solution you obtained is the absolute maximum. A solution that
does not use the method of Lagrange multipliers will receive no credit, even if it
is correct. Note: 32 = 25 .
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