Math 241 §BL1
Problem Set 14
(1) In three variables, the Implicit Function Theorem guarantees that if a differen-
tiable function F(x, y, z) of independent variables xand ysatisfies F(P) = 0 and
DzF(P)6= 0, then z=z(x, y) near P, i.e. we can write zas a function of xand
ynear P.
(a) Use the Chain Rule to show that
∂F
∂x +∂F
∂z
∂z
∂x = 0
and similarly for y.
(b) Use part (a) to compute ∂z
∂x and ∂z
∂y when
(i) xexy +yz +zex= 14
(ii) y2zex+y−sin (xyz ) = 0
(iii) xyz = cos(x+y+z)
(iv) ln(x+yz) = 1 + xy2z3.
(2) Consider (for real numbers a, b) the function f(x, y) = 1
2(ax2+by2).
(a) Show that the origin is a critical point of f(look up the definition in the
text).
(b) For what pair (a, b) does fhave a maximum at the origin? [Use the curves
along the x-axis and y-axis on the graph to get some idea of what’s going
on!] Describe the graph of fin this case.
(c) For what pair (a, b) does fhave a minimum at the origin? Describe the
graph of fin this case.
(3) Find all critical points of the following functions
(a) f(x, y) = 6xy2−2y3−3y4
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