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QUIZ 4, Version A : MATH 251, Section 506
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Write up your result, detail your calculations if necessary and BOX your final answer
Z
1Z x
1. [25pts] Compute the integral
0
sin(x2 )dy dx.
0
2. Let f (x, y) = 4x2 + y 2 − 4x + 2y
(a) [25pts] Find the critical points of this function.
(b) [50pts] Classify these points ( say if these are maximum, minimum points or saddle points by
justifying).
1. We integer the continuous function f (x, y) = sin(x2 ) on {(x, y) ∈ R2 | 0 ≤ x ≤ 1, 0 ≤ y ≤ x}.
Z
0
1
Z
(
x
Z
2
sin(x )dy) dx =
0
1
x
y sin(x2 ) 0 dx,
0
Z
=
0
=
1
cos(x2 )
x sin(x )dx = −
2
2
1
,
0
1 − cos 1
.
2
2. f is a polynomial function. It admits second partial derivatives at any points in R2 .
(a) Determine the critical points of f i.e the sets of points (x, y) such that fx (x, y) = 8x − 4 = 0 =
fy (x, y) = 2y + 2. Solve the following system
8x − 4 = 0
⇔
2y + 2 = 0
(
1
2
y = −1,
x =
1
We obtain the only one critical point : ( , −1) .
2
(b) Now, classify this point. We have fxx (x, y) = 8, fyy (x, y) = 2, fxy (x, y) = 0 = fyx (x, y) (by the
Clairaut’s Theorem since the function is polynomial).
2
= 16 > 0 and fxx ( 21 , −1) = 8 > 0 so ( 12 , −1) is
So, fxx ( 12 , −1)fyy ( 21 , −1) − fxy ( 12 , −1)
a local minimum point.