This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = x2 + y2 + z2; x4 + y4 + z4 = 7

The maximum value is +√21 and the minimum value is -√21

Step-by-step explanation:

f(x,y,z) = x² + y² + z². Let g(x,y,z) = x⁴ + y⁴ + z⁴ - 7 = 0

Using Lagrange multipliers,

df/dx = λg/dx, df/dy = λg/dy. and df/dz = λg/dz

df/dx = 2x, df/dy = 2y, df/dz = 2z

dg/dx = 4x³, dg/dy = 4y³, dg/dz = 4z³

So, df/dx = λg/dx ⇒ 2x = 4λx³     (1)

df/dy = λg/dy ⇒ 2y = 4λy³           (2)

df/dz = λg/dz ⇒ 2z = 4λz³           (3)

From (1) 4λx³ - 2x = 0

2λx³ - x = 0

x(2λx² - 1) = 0

solving, x = 0 or (2λx² - 1) = 0 ⇒ 2λx² = 1 ⇒ x = ±1/√(2λ) since x ≠ 0

From (2) 4λy³ - 2y = 0

2λy³ - y = 0

y(2λy² - 1) = 0

solving, y = 0 or (2λy² - 1) = 0 ⇒ 2λy² = 1 ⇒ y = ±1/√(2λ) since y ≠ 0

From (3) 4λz³ - 2z = 0

2λz³ - z = 0

z(2λz² - 1) = 0

solving, z = 0 or (2λz² - 1) = 0 ⇒ 2λz² = 1 ⇒ z = ±1/√(2λ) since z ≠ 0

g(x,y,z) = x⁴ + y⁴ + z⁴ - 7 = 0

(1/√(2λ))⁴ + (1/√(2λ))⁴ + (1/√(2λ))⁴ - 7 = 0

3 (1/√(2λ))⁴ = 7

(1/√(2λ))⁴ = 7/3

1/√(2λ) = ⁴√7/3

√(2λ) = ⁴√3/7

2λ = √3/7

λ = 1/2(√3/7)

Since x = 1/√(2λ) = 1/√(2 [1/2(√3/7)]) = 1/⁴√3/7 = ±⁴√7/3

Also y = 1/√(2λ) = 1/√(2 [1/2(√3/7)]) = 1/⁴√3/7 = ±⁴√7/3

and z = 1/√(2λ) = 1/√(2 [1/2(√3/7)]) = 1/⁴√3/7 = ±⁴√7/3

Substituting x,y and z into f(x,y,z) we have

f(x,y,z) = (⁴√7/3)² + (⁴√7/3)² + (⁴√7/3)² = 3(⁴√7/3)² = 3(√7/3) = √(7 × 3) = ±√21

The maximum value is +√21 and the minimum value is -√21

hey there!

if she's saving for six months, let's divide the total number we need for the trip by six, dividing it into the amount that needs to be saved for each month.

500 ÷ 6 =

she needs to save at least this amount each month, so we should use this symbol: ≥

let x be the amount she saves each month.

your final answer is x ≥

hope this !

answer: its c because when it has a magnet next to it it will move away from the mixture. hope this

step-by-step explanation: