In a box of chocolate candies, 25 of the chocolates have peanuts. 6 of the chocolates have peanuts.

Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Allushta [10]

Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
$f_x=5yz=0\implies y=0\text{ or }z=0$
$f_y=5xz=0\implies x=0\text{ or }z=0$
$f_z=5xy=0\implies x=0\text{ or }y=0$

Taken together, we find that (0, 0, 0) appears to be the only critical point on $f$ within the ball. At this point, we have $f(0,0,0)=0$ .

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

$L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)$

with partial derivatives (set to 0)

$L_x=5yz+2\lambda x=0$
$L_y=5xz+2\lambda y=0$
$L_z=5xy+2\lambda z=0$
$L_\lambda=x^2+y^2+z^2-1=0$

We then observe that

$xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2$

So, ignoring the critical point we've already found at (0, 0, 0),

$5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}$
$5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}$
$5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}$

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of $\left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right)$ , at which points we get a value of either of $\pm\dfrac5{\sqrt3}$ , with the maximum being the positive value and the minimum being the negative one.

5 0

3 years ago

graph the line that passes through the points (-8,9) and (4,-3) what are the x and y intercepts for this line someone help pleas

VashaNatasha [74]

Answer:

(0,1)- y intercept (1,0) x intercept

Step-by-step explanation:

first lets write this equation in slope intercept form: y=mx+b

m=slope

b=y intercept

so

slope formula:

(y2-y1)/(x2-x1)

y2=-3

y1=9

x2=4

x1=-8

so

(-3-9)/(4--8)=-12/12=-1

slope=-1

hence

y=-1x+b

substitute "x" as 4 and "y" as -3

so

-3=-1(4)+b

-3=-4+b

1=b

so

y=-x+1

1=y intercept

and

0=-x+1

-1=-x

1=x

1=x intercept

4 0

3 years ago

The median price, P, of a home rose from 80$ thousand in 1980 to 240$ thousand in 2000. Let t be the number of years since 1980.

Elena L [17]

Answer:

1

Step-by-step explanation:

2

3

4

5

6

7

8 0

2 years ago

Does anyone know what the slope should be

Lemur [1.5K]

The slope is 4/3.

rise/run
rise = 4
run = 3

5 0

2 years ago

Blake has b baseball cards. His aunt gave him more cards, increasing his total number of cards by 25%. Which expression represen

Vera_Pavlovna [14]

.25b+ b is the answer to ur question

6 0

3 years ago

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