TUTORIAL - Mathematics - 2013/2014
Session n. 6
Implicit function theorem
Exercise 1. Show that the equation 3x2 − 3xy 2 + y 3 + 3y 2 = 4 defines y implicitly as a function h(x)
of x in a neighborhood of the point (1, 1). Moreover
a) compute the equation of the line tangent to the graph of h(x) at x = 1.
b) [optional -homework ] compute h00 (x) and say whether, at a neighborhood of x = 1, h is (locally)
convex or concave.
c) [important] Now assume F (x, y) = 3x2 − 3xy 2 + y 3 + 3y 2 represents the production function of a
certain good, with x and y the quantities of the two necessary inputs. Assume that x decreases to 0.9:
what has to be the value of y in order to have, approximately, the same quantity F (1, 1) of output?
Exercise 2. The equation
x3 ln x + y 3 ln y = 2z 3 ln z
defines z as a differentiable function h(x, y) of x and y in a neighborhood of the point (x, y, z) = (e, e, e).
Calculate h0x (e, e) and h0y (e, e), and the equation of the plane tangent to the graph of h at (e, e, e).
Optimization
Exercise 3. Determine the global maximum and minimum of the function
f (x, y, z) = x − 2y + 2z
over the set
A := {(x, y, z) ∈ R3 , x2 + y 2 + z 2 = 9}.