Flat points: peaks, pits, and saddles

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

For f(x, y) = x squared minus 2x + y squared + 6y + 10, what is the critical point?
- (1, 3)
- (minus 1, minus 3)
- (1, minus 3)
Jay says a negative discriminant D means a saddle point. Is Jay right?
Circle one: True False
For f(x, y) = x squared + y squared, what is D = fxx fyy minus fxy squared?
Answer: ______________
Jay says a negative discriminant D means a saddle point. Is Jay right?
Circle one: True False
For f(x, y) = x^2 + y^2, what is D = fxx fyy - fxy^2?
Answer: ______________
Amara says f(x, y) = x^2 - y^2 has a saddle at (0, 0). Is Amara right?
Circle one: True False
For f(x, y) = x^2 - 2x + y^2 + 6y + 10, what is the critical point?
- (1, -3)
- (1, 3)
- (-1, -3)
- (0, 0)
How does f(x, y) = x^2 + y^2 behave at (0, 0)?
- local minimum
- local maximum
- saddle
- no critical point
Why can one variable tests miss a saddle?
- they only check axis slices, which can hide the up-down mix
- they compute D with the wrong sign
- they assume every critical point is a max
- they require three variables
Why can one-variable tests miss a saddle?
- they compute D with the wrong sign
- they only check axis slices, which can hide the up-down mix
- they assume every critical point is a max

Answer key

For grown-ups. Fold this page away before handing over the rest.

(1, minus 3) · fx = 2x minus 2 = 0 gives x = 1; fy = 2y + 6 = 0 gives y = minus 3.
True · Jay is right: D < 0 is exactly the saddle case of the test.
4 · fxx = 2, fyy = 2, fxy = 0, so D = 4 minus 0 = 4.
True · Jay is right: D < 0 is exactly the saddle case of the test.
4 · fxx = 2, fyy = 2, fxy = 0, so D = 4 - 0 = 4.
True · Amara is right: D = minus 4 < 0 means a saddle.
(1, -3) · fx = 2x - 2 = 0 gives x = 1; fy = 2y + 6 = 0 gives y = minus 3.
local minimum · Gradient zero with D = 4 > 0 and fxx = 2 > 0 means a local minimum.
they only check axis slices, which can hide the up-down mix · Axis slices of x^2 - y^2 can each look fine while the full surface rises one way and falls the other.
they only check axis slices, which can hide the up-down mix · Axis slices of x squared minus y squared can each look fine while the full surface rises one way and falls the other.