If the plane `2x-y+2z+3=0` has the distances `(1)/(3)` and `(2)/(3)` units from the planes `4x-2y+4z+lambda=0` and `2x-y+2z+mu=0`, respectively, then the maximum value of `lambda+mu` is equal to

Explanation

To solve the problem, we need to find the maximum value of \( \lambda + \mu \) given the distances from the plane \( 2x - y + 2z + 3 = 0 \) to the other two planes.

### Step 1: Identify the planes and their equations
We have the following planes:
1. Plane \( P: 2x - y + 2z + 3 = 0 \)
2. Plane \( P_1: 4x - 2y + 4z + \lambda = 0 \)
3. Plane \( P_2: 2x - y + 2z + \mu = 0 \)

### Step 2: Calculate the distance from plane \( P \) to plane \( P_1 \)
The distance \( d \) from a plane \( Ax + By + Cz + D = 0 \) to another plane \( Ax + By + Cz + D' = 0 \) is given by the formula:

\[ d = \frac{|D' - D|}{\sqrt{A^2 + B^2 + C^2}} \]

For the distance from \( P \) to \( P_1 \):
- \( A = 4 \), \( B = -2 \), \( C = 4 \)
- \( D = 3 \) (from plane \( P \))
- \( D' = \lambda \) (from plane \( P_1 \))

The distance is given as \( \frac{1}{3} \):

\[ \frac{|\lambda - 3|}{\sqrt{4^2 + (-2)^2 + 4^2}} = \frac{1}{3} \]

Calculating the denominator:

\[ \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \]

Thus, we have:

\[ \frac{|\lambda - 3|}{6} = \frac{1}{3} \]

### Step 3: Solve for \( \lambda \)
Cross-multiplying gives:

\[ |\lambda - 3| = 2 \]

This results in two cases:

1. \( \lambda - 3 = 2 \) → \( \lambda = 5 \)
2. \( \lambda - 3 = -2 \) → \( \lambda = 1 \)

### Step 4: Calculate the distance from plane \( P \) to plane \( P_2 \)
For the distance from \( P \) to \( P_2 \):
- The coefficients are the same as plane \( P \) (since \( P_2 \) has the same coefficients for \( x, y, z \)).
- The distance is given as \( \frac{2}{3} \):

\[ \frac{|\mu - 3|}{6} = \frac{2}{3} \]

Cross-multiplying gives:

\[ |\mu - 3| = 4 \]

This results in two cases:

1. \( \mu - 3 = 4 \) → \( \mu = 7 \)
2. \( \mu - 3 = -4 \) → \( \mu = -1 \)

### Step 5: Calculate \( \lambda + \mu \)
Now we have the possible values for \( \lambda \) and \( \mu \):
- \( \lambda = 5 \) or \( 1 \)
- \( \mu = 7 \) or \( -1 \)

Calculating \( \lambda + \mu \) for all combinations:
1. \( \lambda = 5, \mu = 7 \) → \( \lambda + \mu = 12 \)
2. \( \lambda = 5, \mu = -1 \) → \( \lambda + \mu = 4 \)
3. \( \lambda = 1, \mu = 7 \) → \( \lambda + \mu = 8 \)
4. \( \lambda = 1, \mu = -1 \) → \( \lambda + \mu = 0 \)

### Step 6: Find the maximum value
The maximum value of \( \lambda + \mu \) is:

\[ \max(12, 4, 8, 0) = 12 \]

### Final Answer
The maximum value of \( \lambda + \mu \) is \( 12 \).