Mark P Neyer
Questions for Final Exam
1. The following is a graph of the level curve of a function f(x,y), as well as two
constraint paths. In the region shown, the gradient of f is always defined and is
never the zero vector. At which points do constrained local maxima or minima
occur?
a) All three points are local maxima or minima because the graph of the constraint
touches the graph of the level curve. Points 1 and 3 are maxima because they
cross the level curve, whereas point 2 is a minimum because it does not cross the
level curve.
b) There is a local maximum or minimum at point 2, because the constraint curve
is parallel to the level curve.
c) There is a local maximum at point 2, because the constraint curve is parallel to
the level curve, and the constraint curve decreases on either side of point 2.
d) There is no way to tell.
Answer: b. There is a local minimum or maximum at point 2, because the
constraint curve is parallel to the level curve of f. There is a constrained minimum
or maximum on f whenever the constraint is parallel to the level curve of f. We
don't know whether or not it is a local minimum or maximum because we don't
have any information about which side of the level curve is higher, so c is
incorrect.
2. If we want to optimize f(x,y) subject to the constraint g(x,y) = 0, we solve the
equation the gradient of f(x,y) = λ the gradient of g(x,y). What does λ represent?
a) It represents the slope of the level curve of f.
b) It represents the slope of the constraint curve g.
c) It doesn't really represent anything; it ensures that f(x,y) is parallel to the
gradient of g(x,y).
d) It represents the ratio of the constraint to the level curve.
Answer: c. It doesn't really “represent” anything on the graph, λ is just a
placeholder to show that the gradient of f(x,y) is parallel to the gradient of g(x,y),
because if two vectors are parallel, then one of them has to be representable as
some constant times the other.
3. The following is a graph of the level curve of a function f(x,y), and a constraint
curve g(x,y). Is there a constrained local minimum or maximum of f at the point
of intersection?
a) There is no way to tell.
b) There is either a maximum or minimum at the point of intersection because the
lines cross each other.
c) There not a maximum or minimum at the point of intersection because the level
curve is not parallel to the constraint curve.
d) There is not a maximum because the two curves are not perpendicular, but
there could be a minimum.
Answer: a. There is no way to tell because we were not told if the gradient of f
was the zero vector at the point of intersection. The fact that they cross tells us if
the gradient of f is not the zero vector then there isn't a minimum or maximum,
but it tells us nothing if the gradient off is zero.
4. Homsar is trying to find constrained local maxima and minima for function f(x,y),
with constraint equation g(x,y) = 0. He set up the equation “the gradient of f(x,y)
= λ the gradient of g(x,y)” and tried to find an x and y that fit, but he keeps getting
the result -1=5. He says he's checked his math repeatedly and swears he hasn't
made any mistakes. What advice can you give him?
a) Parameterize the constraint equation, plug the result into f, and then take the
derivative of that function and look for where it is equal to zero, checking your
endpoints as well. Seeing it in a new way will keep you from making the same
mistake repeatedly.
b) Ask Strong Sad to do it for you.
c) Try switching the equations to polar coordinates – things might get easier for
you.
d) Maybe there are no constrained local minima or maxima to find.
Answer: d. There aren't always constrained minima or maxima, and if you try to
solve “the gradient of f(x,y) = λ the gradient of g(x,y)” when this is the case,
you'll get a screwy result like -1=5.