As x increases, which function will eventually have the greatest outputs?

Find the extreme values of f subject to both constraints f(x,y) = 2x^2+3y^2-4x-5, x^2 + y^2 <=16

Softa [21]

$f(x,y)=2x^2+3y^2-4x-5$
$f_x=4x-4=0\implies x=1$
$f_y=6y=0\implies y=0$

$f(x,y)$ has only one critical point at $(x,y)=(1,0)$ . The function has Hessian

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}4&0\\0&6\end{bmatrix}$

which is positive definite for all $(x,y)$ , which means $f(x,y)$ attains a minimum at the critical point with a value of $f(1,0)=-7$ .

To find the extrema (if any) along the boundary, parameterize it by $x=4\cos t$ and $y=4\sin t$ , with $0\le t$ . On the boundary, we have

$f(x(t),y(t))=F(t)=2(4\cos t)^2+3(4\sin t)^2-4(4\cos t)-5=32\cos^2t+48\sin^2t-16\cos t-5$
$F(t)=35-16\cos t-8\cos2t$

Find the critical points along the boundary:

$F'(t)=16\sin t+16\sin2t=16\sin t+32\sin t\cos t=16\sin t(1+2\cos t)=0$
$\implies t=0,\dfrac{2\pi}3,\pi,\dfrac{4\pi}3$

Respectively, plugging these values into $F(t)$ gives 11, 47, 43, and 47. We omit the first and third, as we can see the absolute extrema occur when $F(t)=47$ .

Now, solve for $x,y$ for both cases:

$t=\dfrac{2\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=2\sqrt3\end{cases}$

$t=\dfrac{4\pi}3\implies\begin{cases}x=4\cos t=-2\\y=4\sin t=-2\sqrt3\end{cases}$

so $f(x,y)$ has two absolute maxima at $(x,y)=(-2,\pm2\sqrt3)$ with the same value of 47.

5 0

3 years ago

Choose the best answer.

ElenaW [278]

Saving Account are those kind of accounts by a single Person or two or more persons in which you can deposit or withdrew money at any instant, either with the help of that Bank in which you have your Account, or ,ATM's or Through Internet Banking.

Among the four options provided

The Type of Account which is not a Savings account

Option D : Automatic Account

First three are different kind of Saving account.

6 0

3 years ago

3/5 and4/7 put these over a common denominator

pishuonlain [190]

The common denominater for the question is 35. Multiply 7 x 5 because in this case, they have no number before that which is common.

8 0

4 years ago

Read 2 more answers

Please help me!!!! I don’t understand

ololo11 [35]

Answer:

Step-by-step explanation:

If ∠EFG and ∠GFH are a linear pair, then m∠EFG + m∠GFH = 180°

(look at the picture).

We have

m∠EFG = 5n + 16

m∠GFH = 2n + 10

Substitute:

(5n + 16) + (2n + 10) = 180 <em>combine like terms</em>

(5n + 2n) + (16 + 10) = 180

7n + 26 = 180 <em>subtract 26 from both sides</em>

7n + 26 - 26 = 180 - 26

7n = 154 <em>divide both sides by 7</em>

7n/7 = 154/7

n = 22

Insert the value of n into expressions that specify angle measures:

m∠EFG = 5n + 16 → m∠EFG = 5(22) + 16 = 110 + 16 = 126°

m∠GFH = 2n + 10 → m∠GFH = 2(22) + 10 = 44 + 10 = 54°

8 0

3 years ago

Homework Progress<br>100 / 285<br>Find the value of x in each of the following<br>a) = 8<br>b) X = 6

SCORPION-xisa [38]

Can you post a picture of the question?

7 0

3 years ago