## Step 1: Define the given functions and constraint

We have the function F(x, y, z) = x^2 + y^2 + z^2 and the constraint G(x, y, z) = (x^2 / 64) + (y^2 / 36) + (z^2 / 25) - 1 = 0.
#Step 2: Introduce the Lagrange multiplier#

## Step 2: Introduce the Lagrange multiplier

To find the extremum values, we use the method of Lagrange multipliers. Define a function L(x, y, z, λ) = F(x, y, z) - λG(x, y, z) where λ is the Lagrange multiplier.
#Step 3: Define the equations to solve for extremum points#

## Step 3: Define the equations to solve for extremum points

Now, we need to solve the system of equations obtained by setting the partial derivatives of L(x, y, z, λ) with respect to x, y, z, and λ equal to zero, as follows:
∂L/∂x = 2x - λ(2x / 64) = 0
∂L/∂y = 2y - λ(2y / 36) = 0
∂L/∂z = 2z - λ(2z / 25) = 0
G(x, y, z) = (x^2 / 64) + (y^2 / 36) + (z^2 / 25) - 1 = 0
#Step 4: Solve the system of equations for extremum points#

## Step 4: Solve the system of equations for extremum points

Solving the first three equations for λ, we get:
λ = 32x / x^2
λ = 18y / y^2
λ = 12z / z^2
Since all values of λ are equal, we can set them equal to each other as follows:
32x / x^2 = 18y / y^2 = 12z / z^2
Now we substitute the values of x, y, and z from the constraint equation into these equations to solve for x, y, and z:
x^2 = 64(18y / y^2) = 64(12z / z^2)
y^2 = 36(32x / x^2) = 36(12z / z^2)
z^2 = 25(32x / x^2) = 25(18y / y^2)
This yields 4 points: (±4, ±3, ±5).
#Step 5: Calculate the function values at extremum points#

## Step 5: Calculate the function values at extremum points

We now find F(4, 3, 5), F(-4, 3, 5), F(4, -3, 5), and F(-4, -3, 5):
F(4, 3, 5) = 16 + 9 + 25 = 50
F(-4, 3, 5) = 16 + 9 + 25 = 50
F(4, -3, 5) = 16 + 9 + 25 = 50
F(-4, -3, 5) = 16 + 9 + 25 = 50
All four extremum points have the same functional value, F = 50.
#Step 6: Determine the maximum and minimum values and points#

## Step 6: Determine the maximum and minimum values and points

The surface of the ellipsoid is closed and bounded, so there must be a maximum and minimum value for F on the surface. Since all four extremum points have the same value, this must be both the maximum and minimum value.
The maximum and minimum value of F(x, y, z) on the surface of the ellipsoid is F = 50, and the points at which these values are achieved are (±4, ±3, ±5).