Homework # 14 - Math 250
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
In questions 1 through 8, let \(f(x) = x^3+3xy^2-3x\), and we will consider the critical points.
1. In the process of finding all critical points, we observe that
a. The curve \(f_x(x,y) = 0\) is a parabola and \(f_y(x,y) = 0\) is a straight line.
b. The curve\(f_x(x,y) = 0\) is a circle and \(f_y(x,y) = 0\) is a straight line.
c. The curve\(f_x(x,y) = 0\) is a circle and \(f_y(x,y) = 0\) is two intersecting straight lines.
d. The curve\(f_x(x,y) = 0\) is a straight line and \(f_y(x,y) = 0\) is a circle.
e. The curve\(f_x(x,y) = 0\) is two intersecting straight lines and \(f_y(x,y) = 0\) is a circle.
2. How many critical points are there?
a. 2
b. 3
c. 4
d. 5
e. 6
3. How many relative maxima are there?
a. 0
b. 1
c. 2
d. 3
e. 4
4. How many relative minima are there?
5. How many saddle points are there?
6. In this problem, for how many critical points does the test fail?
7. The point \((1,0)\) is
a. a relative max.
b. a relative min.
c. a saddle point.
d. a critical point, but the test gives no information about this point.
e. none of the above.
8. The point \((0,0)\) is
In questions 9 through 13, let \(f(x) = x^3 - 6xy + y^3\), and let the region $$R = \{(x,y): x\leq 3, y\leq 3, x+y \geq 3\}$$
9. The region \(R\) is shaped like
a. an open square
b. a closed square
c. an open triangle.
d. a closed triangle.
e. a T-Rex with sharp teeth.
10. The maximum of \(f\) on \(R\) occurs
a. once in the interior.
b. twice in the interior.
c. once on the boundary.
d. twice on the boundary.
e. once in the interior and once on the boundary.
11. The minimum of \(f\) on \(R\) occurs
12. To the nearest one place after the decimal, the minimum value of \(f\) on \(R\) is
a. 0.0
b. -2.4
c. -6.8
d. -8.0
e. -11.2
13. How many of the critical points of \(f\) are outside \(R\)?
e. infinitely many