Quiz 6
Math 1321 - Accelerated Engineering Calc II
Name:
March 11, 2016
Quiz Score:
/10
Answer each question completely in the area below. Show all work and explain your
reasoning. If the work is at all ambiguous, it is considered incorrect. No phones, calculators,
or notes are allowed. Anyone found violating these rules will be asked to leave immediately.
Point values are in the square to the left of the question. If there are any other issues,
please ask the instructor.
7
1. Consider the function
f (x, y ) = 50 − x 2 − 2y 2 .
Find the local extrema (minimima, maxima) of f (x, y ). Make sure to clearly state which type
of extrema you’ve found and why.
Solution: To find the local extrema of a function, we find the critical points, candidates for
extrema, which are characterized by the gradient being zero, that is:
0 = ∇f = hfx , fy i = h−2x, −4y i.
Solving these two equations:
−2x = 0,
−4y = 0
=⇒ x = 0, y = 0,
thus our only critical point is the origin, (0, 0). We can now classify this as a local minimum
or maximum by using the second derivative test, which says we must evaluate:
D = fxx fy y − (fxy )2 = (−2)(−4) − 0 = 8.
Thus, we have fxx = −2 < 0 and D = 8 > 0, which, by the second derivative test, means
this is a local maximum.
1/2
Quiz 6
3
Math 1321 - Accelerated Engineering Calc II
March 11, 2016
2. Can you say anything about the local extrema you found in part (a) being global extrema?
Hint: look at the function and how it behaves as x, y change.
Solution: Although a bounded, closed domain (and a continuous function) always ensures
a global minimum and maximum, sometimes we have a global minimum or maximum even
without a specified domain. Here, we observe that the function is basically
f (x, y ) = 50 − something positive − something positive,
meaning that the largest this function can be is f (x, y ) = 50, when the “something positive”
terms are zero, at our maximum x = 0, y = 0.
y
-5
5
0
40
20
z
0
-20
-5
0
x
5
2/2