Assignment 30

course Mth 272

8/1 5 pm

030.

*********************************************

Question: `qQuery problem 7.5.10 extrema of x^2+6xy+10y^2-4y+4

List the relative extrema and the saddle points of the function and tell which is which, and how you obtained each.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: to find the extrema and saddle points, we must first find the first and second partial derivatives with respect to x and y. Using our derivative rules, we find that F? = 2x+6y and F?x = 2. F? = 6x + 20y -4 and F?y = 20. We also find that F?y = 6. Now we can look for any extrema by setting the first partial derivatives equal to 0 and solving for x and y. After using basic algebra to solve the system of equations, we find that (-6,2) satisfies the expressions. Now we can use the Second Partials Test for relative extrema to find out what kind of extrema (-6,2) is. So (2)(20) ?6)^2 = 40 ?36 = 4. Since d>0 and F?x >0, we have a maximum at (-6,2).

confidence rating: 2

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

`a fx = 2x + 6y ; fy = 6x + 20 y - 4, where fx and fy mean x and y partial derivatives of f.

fxx is the x derivative of fx and is therefore 2

fyy is the y derivative of fy and is therefore 20

fxy is the y derivative of fx and is therefore 6. Note that fyx is the x derivative of fy and is therefore 6. fxy and fyx are always the same, provided the derivatives exist.

fx = 0 and fy = 0 if

2x + 6y = 0 and

6x + 20y - 4 = 0.

This is a system of two equations in two unknowns. You can solve the system by any of several methods. Multiplying the first equation by -3 and adding the resulting equations, for example, gives you 2 y = 4 so that y = 2. Substituting y = 2 into the first equation gives you 2x + 6 * 2 = 0, which has solution x = -6.

Whatever method you use, the solution of this system is x = -6, y = 2.

To determine the nature of the critical point at (-6, 2) you have to look at fxx, fyy and fxy. We have

fxx = 2

fyy = 20

fxy = 6.

This is a system of two equations in two unknowns. You can solve the system by any of several methods. Multiplying the first equation by -3 and adding the resulting equations, for example, gives you 2 y = 4 so that y = 2. Substituting y = 2 into the first equation gives you 2x + 6 * 2 = 0, which has solution x = -6.

Whatever method you use, the solution of this system is x = -6, y = 2.

To determine the nature of the critical point at (-6, 2) you have to evaluate the quantity fxx * fyy - 4 fxy^2.

We have

fxx = 2

fyy = 20

fxy = 6.

So fxx * fyy - 4 fxy^2 = 2 * 20 - 4 * 6^2 = 8.

This quantity is positive, so you have either a maximum or a minimum.

Since fxx and fyy are both positive the graph is concave upward in all directions and the point is a minimum.

The coordinates of the point are (-6, 2, (-6)^2+6 * (-6) * 2 + 10* 2^2 - 4 * 2 + 4 ) = (-6, 2, 0)

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): The given answer has a wrong Second Partials Test and wrong algebra (fxx * fyy - 4 fxy^2 = 2 * 20 - 4 * 6^2 = 8). My answer does not match the given answer, but since the given answer is mathematically incorrect, I believe my answer that I found with the book? Second Partials Test and information about d is correct.

Your calculation for the second partials was correct (there were two very obvious errors in the given solution, the first with the formula and the second with the arithmetic). However note that the conclusion, based on the fact that the second partials test confirms that all derivatives at the point are positive, is the same.

(-6, 2) gives the x and y coordinates of the maximum. The graph of the function is in three dimensions, and the maximum does occur at (-6, 2, (-6)^2+6 * (-6) * 2 + 10* 2^2 - 4 * 2 + 4 ) = (-6, 2, 0).

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `qWhat are the critical points and what equations did you solve to get them?

How did you test each critical point to determine if it is a relative max, relative min or saddle point?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The critical points are (-6,2). We solved F?, F?x, F?y, F?, and F?y to get them. After finding F?x, F?y, and F?y, we used the Second Partials Test for Relative Extrema to find d. To tell if it is a relative max, min, or saddle point, we can look to see if d > 0, which determines a max or min, or if d<0, which makes it a saddle point.

confidence rating: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

*********************************************

Question: `qQuery problem 7.5.28 extrema of x^3+y^3 -3x^2+6y^2+3x+12y+7

List the relative extrema and the saddle points of the function and tell which is which, and how you obtained each.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: To find the extrema, we first need to find F?, F?x, F?y, F?, and F?y using our derivative rules. We find that F? = 3x^2 -6x +3, F?x = 6x -6, and F?y = 0. We also find that F? = 3y^2 + 12y +12, and F?y = 6y+12. Now we can set F? and F? equal to 0. So we find that 3x^2 -6x +3 = 0 and 3y^2 +12y +12 = 0. Solving these algebraically, we get the point(1,-2). Since F?y = 0, we find that the Second Partials Test tells us nothing.

confidence rating: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

`a We have fx = 3 x^2 + 6 x + 3 and fy = 3 y^2 + 12 y + 12.

Factoring we get

fx = 3 ( x^2 - 2x + 1) = 3 ( x-1)^2 and

fy = 3 ( y^2 + 4y + 4) = 3 ( y+2)^2.

So

fx = 0 when 3(x-1)^2 = 0, or x = 1 and

fy = 0 when 3(y+2)^2 = 0 or y = -2.

We get

fxx = 6 x - 6 (the x derivative of fx)

fyy = 6 y + 12 (the y derivative of fy) and

fxy = 0 (the y derivative of fx, which is also equal to fyx, the x derivative of fy)

At the critical point x = 1, y = -2 we get fxx = fyy = 0. So the test for max, min and critical point gives us fxx*fyy - 4 fxy^2 = 0, which is inconclusive--it tells us nothing about max, min or saddle point.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `qAt what point(s) did the second-partials test fail?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: From the previous solution, the Second Partials test fails at the point(1,-2) because F?y = 0.

confidence rating: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

`a At the critical point x = 1, y = -2 we get fxx = fyy = 0. So the test for max, min and critical point gives us fxx*fyy - 4 fxy^2 = 0, which is inconclusive--it tells us nothing about max, min or saddle point.

&#Very good work. Let me know if you have questions. &#