What two factors of -21 add up to -20

Find the maximum and minimum values of the function f(x,y)=2x2+3y2−4x−5 on the domain x2+y2≤100. The maximum value of f(x,y) is:

ryzh [129]

First find the critical points of f :

$f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7$

$\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1$

$\dfrac{\partial f}{\partial y}=6y=0\implies y=0$

so the point (1, 0) is the only critical point, at which we have

$f(1,0)=-7$

Next check for critical points along the boundary, which can be found by converting to polar coordinates:

$f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t$

Find the critical points of g :

$\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0$

$\implies\sin t=0\text{ OR }1+5\cos t=0$

$\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi$

where n is any integer. We get 4 critical points in the interval [0, 2π) at

$t=0\implies f(10,0)=155$

$t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299$

$t=\pi\implies f(-10,0)=235$

$t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299$

So f has a minimum of -7 and a maximum of 299.

4 0

3 years ago

Tony bowled 135 and 145 in his first two games. write and solve a compound inequality to find the possible values for a third ga

stira [4]

The average has to be at least 120 and at most 130
To calculate the average we need the sum of all values divided by the number of values, in this case, three (135, 145 and the third result).
120 ≤ (135 + 145 + n)/3 ≤ 130

In inequalities like this, what we change in one side, must be changed in the othe rside as well.
360 ≤ 280 + n ≤ 390
80 ≤ n ≤ 110

5 0

3 years ago

Can you help me please?

9966 [12]

First, change the two mixed numbers in the expression into an improper fraction: 25/6 + 5/3.
Then, find the Lowest Common Denominator of both fractions, which is 6, and set both denominators equal to that. Remember, whatever you do on one side you must do to the other: 25/6 + 10/6
Add the two together: 25/6 +10/6 = 35/6.
To make it a mixed number again, find how many times 6 goes into 35, which is 5 times, with a remainder of 5. Your answer is 5 5/6

6 0

3 years ago

Please help! Will give brainly, 50 points!! I'm stuck with this question and I don't get it!

pishuonlain [190]

Answer: The answer is x^2 + 4x - 3

Step-by-step explanation:

You just combine like terms, the only ones that can combine is the 6x and -4x. You just subtract them to get the answer of x^2 + 4x - 3.

7 0

3 years ago

Read 2 more answers

A deck of cards contains RED cards numbered 1,2,3,4,5, BLUE cards numbered 1,2,3, and GREEN cards numbered 1,2,3,4,5,6. If a sin

Tatiana [17]

Answer: 1/52

Step-by-step explanation:

A deck of cards contains 52 cards

Red cards are numbered 1,2,3

Blue cards are numbered 1,2,3,4,5,6

Green cards are numbered 1,2

Number of Red card = 3

Number of blue card = 6

Number of Green card = 2

Let Pr(R) = Probability of picking a Red card

Let Pr(B) = probability of picking a blue card

Let Pr(G) = probability of picking a green card

Let Pr(RE) = Probability of picking a red even card

Let Pr(RO) = probability of picking a red odd card

Let Pr(BE) =probability of picking a blue even card

Let Pr(BO) = probability of picking a blue odd card

Let Pr(GE) = probability of picking a green even card

Let Pr(GO) = probability of picking a green odd card

Pr(R) = 3/52

Since we have 3 red cards,

Pr(RE) = 1/3

Pr(RO) =2/3

Pr(B) = 6/52

= 3/26

Since we have 6 blue cards ,

Pr(BE) = 3/6

= 1/2

Pr(BO) = 3/6

= 1/2

Pr (G) = 2/52

= 1/26

Since we have 2 green cards,

Pr(GE) = 1/2

Pr(GO) = 1/2

The probability of picking a Green card and an odd green card is

Pr(G) n pr(GO)

1/26 * 1/2

= 1/52

5 0

3 years ago