How do you solve 10x^2 + x

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

PLEASE HELP! THIS IS DUE TOMORROW! Much appreciated.

denis23 [38]

Answer:

Step-by-step explanation:

1a) angle x and angle y are corresponding angles. Both angles lie on the same side of the transversal. Since the lines are parallel, the angles are equal.

1b) angle x and angle y are interior angles on the same side of the transversal. Since the lines are parallel, the angles are equal supplementary.

1c) angle x and angle y are corresponding angles. Both angles lie on the same side of the transversal. Since the lines are parallel, the angles are equal.

1d) angle x and angle y are alternate interior angles. They are between the parallel lines and alternate sides of the transversal. Since the lines are parallel, the angles are equal.

3 0

3 years ago

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Three teachers equally shared StartFraction 4 Over 5 EndFraction of a flat of paper that was donated to the school. What fractio

zhannawk [14.2K]

Answer:

B. 4/15 of the flat

Step-by-step explanation:

5 0

3 years ago

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An orchard owner wants to plant 64 cherry tree seedlings and 96 peach tree seedlings in rows. Each row is to have the same numbe

-BARSIC- [3]

Answer:

Given:

cherry tree seedlings : 64

peach tree seedlings : 96

64/96 : Simplify this fraction

64 / 8 = 8 / 4 = 2

96 / 8 = 12 / 4 = 3

2 and 3 number of seedlings per row

64 / 2 = 32

96 / 3 = 32

number of rows

The maximum number of rows that the owner can plant is 32 rows with 2 cherry tree seedlings and 3 peach tree seedlings.

5 0

3 years ago

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Your round trip drive to school is 3 7/10 miles.how many miles do you drive to and from school in 5 days?

riadik2000 [5.3K]

Answer:

18 1/2 miles

Step-by-step explanation:

3 7/10 * 5

Change to an improper fraction

(10* 3 + 7)/10 * 5

37/10 *5

Rewriting

37 * 5/10

37/2

Changing back to an mixed number

2 goes into 37 18 times with 1 left over

18 1/2

5 0

3 years ago

How do you solve 10x^2 + x - 2 = 0