7/8,0.98,8/9 least to greatest

Consider the following equation. f(x, y) = e−(x − a)2 − (y − b)2 (a) Find the critical points. (x, y) = a,b (b) Find a and b suc

murzikaleks [220]

a.

$f(x,y)=e^{-(x-a)^2-(y-b)^2}\implies\begin{cases}f_x=-2(x-a)e^{-(x-a)^2-(y-b)^2}\\f_y=-2(y-b)e^{-(x-a)^2-(y-b)^2}\end{cases}$

Critical points occur where $f_x=f_y=0$ . The exponential factor is always positive, so we have

$\begin{cases}-2(x-a)=0\\-2(y-b)=0\end{cases}\implies(x,y)=\boxed{(a,b)}$

b. As the previous answer established, the critical point occurs at (-3, 8) if $\boxed{a=-3}$ and $\boxed{b=8}$ .

c. Check the determinant of the Hessian matrix of $f(x,y)$ :

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}$

The second-order derivatives are

$f_{xx}=(-2+4(x-a)^2)e^{-(x-a)^2-(y-b)^2}$

$f_{xy}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}$

$f_{yx}=4(x-a)(y-b)e^{-(x-a)^2-(y-b)^2}$

$f_{yy}=(-2+4(y-b)^2)e^{-(x-a)^2-(y-b)^2}$

so that the determinant of the Hessian is

$\det\mathbf H(x,y)=f_{xx}f_{yy}-{f_{xy}}^2=\left((4(x-a)^2-2)(4(y-b)^2-2)-16(x-a)^2(y-b)^2\right)e^{-2(x-a)^2-2(y-b)^2}$

$\det\mathbf H(x,y)=(16(x-a)^2(y-b)^2-8(x-a)^2-8(y-b)^2)+4)e^{-2(x-a)^2-2(y-b)^2}$

The sign of the determinant is unchanged by the exponential term so we can ignore it. For $a=x=-3$ and $b=y=8$ , the remaining factor in the determinant has a value of 4, which is positive. At this point we also have

$f_{xx}(-3,8;a=-3,b=8)=-2$

which is negative, and this indicates that (-3, 8) is a local maximum.

8 0

3 years ago

(NEED IN UNDER 30 MINUTES HELP HURRY) The dot plots below show the ages of students belonging to two groups of painting classes:

zhuklara [117]

The group that is most likely to have a lower mean age of painting students is group A.

<h3>Which group is most likely to have a lower mean age?</h3>

A dot plot is graph that is used to display data. The dots in a dot plot represents the frequency of a number.

Looking at the graph, it appears that group A has a lower age distribution. Thus, it is likely that it would have a lower mean.

To learn more about dot plots, please check: brainly.com/question/22746300

#SPJ1

5 0

1 year ago

What transformation transforms (a, b) (a, -b)

Ipatiy [6.2K]

Answer:

see explanation

Step-by-step explanation:

Under a reflection in the x- axis

a point (x, y ) → (x, - y )

We have

(a, b ) → (a, - b ) ← reflection in the x- axis

7 0

3 years ago

Find the perimeter of MNP<br> Question #9.

ss7ja [257]

56 is the answer I just took that

8 0

2 years ago

Kennedy buys a bag that contains 6 small boxes of raisins each box contains 6/7 ounce of raisins how many ounces of raisins does

earnstyle [38]

You could do 6*6, which would be 36.
36, which you then divide by 7, the denominator.
5 and 1/7 is the answer.

6 0

2 years ago