Answer:
Minimum value of f(x, y, z) = (1/3)
Step-by-step explanation:
f(x, y, z) = x⁴ + y⁴ + z⁴
We're to maximize and minimize this function subject to the constraint that
g(x, y, z) = x² + y² + z² = 1
The constraint can be rewritten as
x² + y² + z² - 1 = 0
Using Lagrange multiplier, we then write the equation in Lagrange form
Lagrange function = Function - λ(constraint)
where λ = Lagrange factor, which can be a function of x, y and z
L(x,y,z) = x⁴ + y⁴ + z⁴ - λ(x² + y² + z² - 1)
We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.
(∂L/∂x) = 4x³ - λx = 0
λ = 4x² (eqn 1)
(∂L/∂y) = 4y³ - λy = 0
λ = 4y² (eqn 2)
(∂L/∂z) = 4z³ - λz = 0
λ = 4z² (eqn 3)
(∂L/∂λ) = x² + y² + z² - 1 = 0 (eqn 4)
We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z
4x² = 4y²
4x² - 4y² = 0
(2x - 2y)(2x + 2y) = 0
x = y or x = -y
Also,
4x² = 4z²
4x² - 4z² = 0
(2x - 2z) (2x + 2z) = 0
x = z or x = -z
when x = y, x = z
when x = -y, x = -z
Hence, at the point where the box has maximum and minimal area,
x = y = z
And
x = -y = -z
Putting these into the constraint equation or the solution of the fourth partial derivative,
x² + y² + z² = 1
x = y = z
x² + x² + x² = 1
3x² = 1
x = √(1/3)
x = y = z = √(1/3)
when x = -y = -z
x² + y² + z² = 1
x² + x² + x² = 1
3x² = 1
x = √(1/3)
y = z = -√(1/3)
Inserting these into the function f(x,y,z)
f(x, y, z) = x⁴ + y⁴ + z⁴
We know that the two types of answers for x, y and z both resulting the same quantity
√(1/3)
f(x, y, z) = x⁴ + y⁴ + z⁴
f(x, y, z) = (√(1/3)⁴ + (√(1/3)⁴ + (√(1/3)⁴
f(x, y, z) = 3 × (1/9) = (1/3).
We know this point is a minimum point because when the values of x, y and z at turning points are inserted into the second derivatives, all the answers are positive! Indicating that this points obtained are
S = (1/3)
Hope this Helps!!!
