Georgetown University
MATH 137
Multivariable Calculus Fall 2018
Solutions to Midterm 2
1. Find the critical points of f(x; y) = -x3 + 12x - 4y2 and classify each point as a local maximum,
local minimum, or saddl
...
Georgetown University
MATH 137
Multivariable Calculus Fall 2018
Solutions to Midterm 2
1. Find the critical points of f(x; y) = -x3 + 12x - 4y2 and classify each point as a local maximum,
local minimum, or saddle point.
To solve for the critical points, we set
fx
= -3x2 + 12 = 0
fy
= -8y = 0
The first equation gives x2 = 4, whence x = ±2. The second equation gives y = 0.
Thus there are two critical points, (2; 0) and (-2; 0). We now use the Hessian
H = fxxfyy - (fxy)2
to determine the nature of each critical point. We have
) - 02 = 48x.
We have
Therefore f(2; 0) is a local maximum. Also,
H(-2; 0) = 48(-2) = -96 < 0
So (-2; 0; f(-2; 0)) is a saddle point.
2. Use the Lagrange multiplier method to find the maximum and minimum values of
subject to the constraint
We have
We may deduce from the first four equations of this system that x = y = z = t. Hence x2 + y2 +
z2 + t2 = 4x2 = 1, which implies x2 = 1 4, or x = ±12.
Therefore there are two critical points to consider, namely 1 2; 1 2; 12; 12 and -1 2; -1 2; -1 2; -1 2 .
We have
Therefore, on the hypersphere x2 + y2 + z2 + t2 = 1, the absolute maximum of f is 2 and the
absolute minimum of f is -2.
3. Consider the surface defined by the equation x3 + x2y2 + z2 = 0. Find an equation for the plane
tangent to the surface at the point P(-2; 1; 2).
This surface is a level surface of the function f(x; y; z) = x3 +x2y2 +z2. Consequently an equation
of the tangent plane at P(-2; 1; 2) is
Therefore the tangent plane is given by
(a) Find the directional derivative of f at P (1; 1) in the direction of Q(2; 3).
(b) Determine the direction of maximum increase of f at P (1; 1).
This direction is specified by the gradient vector rf(1; 1) = h2; 6i.
5. Show that the level curves f(x; y) = x2 - y2 = k1 and g(x; y) = xy = k2, where k1 and k2 are
constants, intersect orthogonally at any point of intersection P (a; b).
Let P (a; b) be any point of intersection of two level curves. We have
Therefore the level curves intersect orthogonally.
6. Let f(x; y) satisfy rf(x; y) = chx; yi for some constant c and all (x; y) 2 R2. Show that f is
constant on any circle of radius a > 0 centered at the origin.
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