Find the maximum and minimum values of f(x,y,z)=3x+1y+5z on the sphere x^2+y^2+z^2=1
Use Lagrange multipliers to maximize g(x, y, z, ë) = 3x+1y+5z + ë(x^2+y^2+z^2-1) by differentiating it with respect to x, y, z and ë:
dg/dx = 3 + 2ëx = 0;
dg/dy = 1 + 2ëy = 0;
dg/dz = 5 + 2ëz = 0;
dg/dë = x^2+y^2+z^2-1 = 0
so
(a) 2ëx = -3
(b) 2ëy = -1
(c) 2ëz = -5
and (d) x^2+y^2+z^2 = 1
Divide (a) by (b) to get x/y = 3, or x = 3y
Divide (c) by (b) to get z/y = 5, or z = 5y
You can now express (d) entirely in terms of y^2: the rest should be plain sailing! Don't forget that square roots can take either positive or negative values.
Oops - that "ë" is supposed to be a lambda symbol. Wikipedia on "Lagrange Multipliers" will show how it works.
To find the maximum and minimum values of the function f(x,y,z) = 3x + 1y + 5z on the sphere x^2 + y^2 + z^2 = 1, we can use the method of Lagrange multipliers.
Step 1: Set up the equations
We want to find the values of x, y, z, and λ that satisfy the following equations:
1. ∇f(x, y, z) = λ∇g(x, y, z)
2. g(x, y, z) = 0
where ∇f(x, y, z) is the gradient of f(x, y, z), ∇g(x, y, z) is the gradient of g(x, y, z), and λ is the Lagrange multiplier. In this case, f(x, y, z) = 3x + y + 5z and g(x, y, z) = x^2 + y^2 + z^2 - 1.
Step 2: Calculate the gradients
∇f(x, y, z) = (3, 1, 5)
∇g(x, y, z) = (2x, 2y, 2z)
Step 3: Set up the equations
Setting up the equations, we have:
(3, 1, 5) = λ(2x, 2y, 2z)
x^2 + y^2 + z^2 = 1
Step 4: Solve the equations
From the first equation, we get:
3 = 2λx
1 = 2λy
5 = 2λz
Dividing the first two equations, we get:
3/1 = (2λx)/(2λy)
3 = x/y
Substituting this value into the third equation, we have:
5 = 2λz
From the third equation, we get:
2.5 = λz
Substituting these values into the second equation, we have:
1 = 2λy
1 = 2λ(x/(3/y))
1 = 2λ(x/(3/(x/y)))
1 = 2λ((x^2)/3)
From the first equation, we can solve for x:
3 = 2λx
x = (3/2λ)
Substituting this back into the equation we derived from the second equation, we have:
1 = 2λ((3/2λ)^2/3)
1 = (3/2λ)
From this equation, we can solve for λ:
1 = (3/2λ)
(2λ) = 3
λ = 3/2
Substituting this value of λ back into the equation we derived from the third equation, we have:
5 = 2(3/2)(z)
5 = 3z
z = 5/3
Now we can find the values of x and y using the equations we derived earlier:
x = (3/2λ) = (3/2(3/2)) = 1/2
y = (x^2)/3 = ((1/2)^2)/3 = 1/12
Step 5: Evaluate the function at the critical points
To find the maximum and minimum values of f(x, y, z) = 3x + y + 5z, we substitute the values of x, y, and z we obtained into the function:
f(1/2, 1/12, 5/3) = 3(1/2) + (1/12) + 5(5/3) = 3/2 + 1/12 + 25/3 = 14/12 + 1/12 + 100/12 = 115/12
Therefore, the maximum value of f(x, y, z) on the sphere x^2 + y^2 + z^2 = 1 is 115/12.
Similarly, we can evaluate the function at the other critical point:
f(-1/2, -1/12, -5/3) = 3(-1/2) + (-1/12) + 5(-5/3) = -3/2 - 1/12 - 25/3 = -14/12 - 1/12 - 100/12 = -115/12
Therefore, the minimum value of f(x, y, z) on the sphere x^2 + y^2 + z^2 = 1 is -115/12.
Hence, the maximum value is 115/12 and the minimum value is -115/12 of the function f(x,y,z) = 3x + 1y + 5z on the sphere x^2+y^2+z^2=1.
To find the maximum and minimum values of the function f(x, y, z) = 3x + 1y + 5z on the sphere x^2 + y^2 + z^2 = 1, we can utilize a common technique known as the Lagrange multipliers method.
Step 1: Formulate the Lagrange function:
Start by defining the Lagrange function L(x, y, z, λ) as follows:
L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 1),
where g(x, y, z) = x^2 + y^2 + z^2 - 1 is the equation of the sphere.
Step 2: Calculate the partial derivatives:
Compute the partial derivatives of L(x, y, z, λ) with respect to x, y, z, and λ. Set each derivative equal to zero. We obtain the following equations:
∂L/∂x = 3 - 2λx = 0,
∂L/∂y = 1 - 2λy = 0,
∂L/∂z = 5 - 2λz = 0,
∂L/∂λ = x^2 + y^2 + z^2 - 1 = 0.
Step 3: Solve the system of equations:
Solve the system of equations simultaneously to find the values of x, y, z, and λ.
From ∂L/∂x = 3 - 2λx = 0, we get 2λx = 3, which implies x = 3/(2λ).
From ∂L/∂y = 1 - 2λy = 0, we obtain 2λy = 1, resulting in y = 1/(2λ).
From ∂L/∂z = 5 - 2λz = 0, we have 2λz = 5, yielding z = 5/(2λ).
Substituting these values into the equation ∂L/∂λ = x^2 + y^2 + z^2 - 1 = 0, we can solve for λ.
(x^2 + y^2 + z^2 - 1) = (3/(2λ))^2 + (1/(2λ))^2 + (5/(2λ))^2 - 1 = 0.
Simplifying the equation and finding a common denominator gives:
9/4λ^2 + 1/4λ^2 + 25/4λ^2 = 1.
Combining like terms results in:
35/4λ^2 = 1.
Multiplying both sides by 4/35 gives:
λ^2 = 4/35.
Taking the square root of both sides, we find two potential values for λ:
λ = ± 2/√35.
Step 4: Find x, y, and z:
Substitute the values of λ into the equations x = 3/(2λ), y = 1/(2λ), and z = 5/(2λ) to calculate the corresponding values of x, y, and z.
When λ = 2/√35, we get:
x = 3/(2 * 2/√35) = 3√35/4,
y = 1/(2 * 2/√35) = √35/4,
z = 5/(2 * 2/√35) = 5√35/4.
When λ = -2/√35, we have:
x = 3/(2 * -2/√35) = -3√35/4,
y = 1/(2 * -2/√35) = -√35/4,
z = 5/(2 * -2/√35) = -5√35/4.
Step 5: Substitute the values of x, y, and z into f(x, y, z):
Substitute the values of x, y, and z calculated in step 4 into the function f(x, y, z) = 3x + y + 5z. This will give us the maximum and minimum values of f(x, y, z) on the sphere.
For the value λ = 2/√35:
f(x, y, z) = 3(3√35/4) + (√35/4) + 5(5√35/4) = (9√35 + √35 + 25√35)/4 = (35√35)/4.
For the value λ = -2/√35:
f(x, y, z) = 3(-3√35/4) - (√35/4) + 5(-5√35/4) = (-9√35 - √35 - 25√35)/4 = (-35√35)/4.
Therefore, the maximum value of f(x, y, z) = 3x + y + 5z on the sphere x^2 + y^2 + z^2 = 1 is (35√35)/4, and the minimum value is (-35√35)/4.