Suppose p,q,r and real number such that `q=p(4-p),r=q(4-q),p=r(4-r)`. The maximum possible value of p+q+r is

Options

A. 0
B. 3
C. 9
D. 27

Explanation

To solve the problem, we need to find the maximum possible value of \( p + q + r \) given the relationships between \( p, q, \) and \( r \).

### Step-by-Step Solution:

1. **Understanding the Relationships**:
We have the following equations:
\[ q = p(4 - p) \]
\[ r = q(4 - q) \]
\[ p = r(4 - r) \]

2. **Expressing \( p + q + r \)**:
We want to express \( p + q + r \) in terms of \( p \) alone. First, substitute \( q \) into the equation for \( r \):
\[ r = (p(4 - p))(4 - p(4 - p)) \]

3. **Simplifying \( r \)**:
We need to simplify \( r \):
\[ r = p(4 - p)(4 - 4p + p^2) \]
This expression can be further simplified, but let's keep it in this form for now.

4. **Finding \( p + q + r \)**:
Now, we can express \( p + q + r \):
\[ p + q + r = p + p(4 - p) + r \]
Substitute \( r \) from the previous step:
\[ p + q + r = p + p(4 - p) + p(4 - p)(4 - 4p + p^2) \]

5. **Maximizing \( p + q + r \)**:
To find the maximum value, we can use calculus or symmetry. We can also observe that if \( p, q, r \) are all equal, we can set \( p = q = r = x \). Then we have:
\[ x = x(4 - x) \]
This simplifies to:
\[ x(4 - x) = x \implies 4x - x^2 = x \implies x^2 - 3x = 0 \implies x(x - 3) = 0 \]
Thus, \( x = 0 \) or \( x = 3 \).

6. **Calculating \( p + q + r \)**:
If \( p = q = r = 3 \):
\[ p + q + r = 3 + 3 + 3 = 9 \]

7. **Conclusion**:
The maximum possible value of \( p + q + r \) is \( 9 \).