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3 years ago

Help please 10 points

Mathematics

1 answer:

-BARSIC- [3]3 years ago

6 0

the answer is A

im doing this too ;P

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Use Lagrange multipliers to find the maximum and minimum values of (i) f(x,y)-81x^2+y^2 subject to the constraint 4x^2+y^2=9. (i

sp2606 [1]

i. The Lagrangian is

$L(x,y,\lambda)=81x^2+y^2+\lambda(4x^2+y^2-9)$

with critical points whenever

$L_x=162x+8\lambda x=0\implies2x(81+4\lambda)=0\implies x=0\text{ or }\lambda=-\dfrac{81}4$

$L_y=2y+2\lambda y=0\implies2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1$

$L_\lambda=4x^2+y^2-9=0$

If $x=0$ , then $L_\lambda=0\implies y=\pm3$ .
If $y=0$ , then $L_\lambda=0\implies x=\pm\dfrac32$ .
Either value of $\lambda$ found above requires that either $x=0$ or $y=0$ , so we get the same critical points as in the previous two cases.

We have $f(0,-3)=9$ , $f(0,3)=9$ , $f\left(-\dfrac32,0\right)=\dfrac{729}4=182.25$ , and $f\left(\dfrac32,0\right)=\dfrac{729}4$ , so $f$ has a minimum value of 9 and a maximum value of 182.25.

ii. The Lagrangian is

$L(x,y,z,\lambda)=y^2-10z+\lambda(x^2+y^2+z^2-36)$

with critical points whenever

$L_x=2\lambda x=0\implies x=0$ (because we assume $\lambda\neq0$ )

$L_y=2y+2\lambda y=0\implies 2y(1+\lambda)=0\implies y=0\text{ or }\lambda=-1$

$L_z=-10+2\lambda z=0\implies z=\dfrac5\lambda$

$L_\lambda=x^2+y^2+z^2-36=0$

If $x=y=0$ , then $L_\lambda=0\implies z=\pm6$ .
If $\lambda=-1$ , then $z=-5$ , and with $x=0$ we have $L_\lambda=0\implies y=\pm\sqrt{11}$ .

We have $f(0,0,-6)=60$ , $f(0,0,6)=-60$ , $f(0,-\sqrt{11},-5)=61$ , and $f(0,\sqrt{11},-5)=61$ . So $f$ has a maximum value of 61 and a minimum value of -60.

5 0

3 years ago

A triangle has points R(-2,0) ,S(2,8) and T (8,2) as shown in the group. M is the midpoint of RT . What is the length of SM

wlad13 [49]

The midpoint is calculated by averaging the coordinates of its endpoints:

$M = \left(\dfrac{-2+8}{2},\dfrac{0+2}{2}\right)=(3,1)$

Now we can use the usual distance formula to get the length of SM:

$SM=\sqrt{(2-3)^2+(8-1)^2}=\sqrt{1+49}=\sqrt{50}$

5 0

3 years ago

Each expression with the appropiate property:

Lisa [10]

Answer:

it denotes existence of additive inverse i.e. option A is correct

Explanation:

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive.

5 0

3 years ago

Tree A has a bright 8h-1 feet. Tree B has a height of 6h-11 feet. How much taller in feet is Tree A than Tree B?

Romashka-Z-Leto [24]

That is a projectile motion problem

6 0

3 years ago

Write the equation of the line that passes through the point (-4,-3) and has a slope of 5?

Viktor [21]

Answer:

y + 3 = 5 (x + 4) or y = 5x + 17

Step-by-step explanation:

7 0

3 years ago