If `a^2 + b = 2`, then maximum value of the term independent of x in the expansion of `( ax^(1/6) + bx^(-1/3))^9` is `(a > 0; b > 0)`

Options

A. 48
B. 84
C. 42
D. 168

Answer

To find the maximum value of the term independent of \( x \) in the expansion of \( (ax^{1/6} + bx^{-1/3})^9 \), given that \( a^2 + b = 2 \) and \( a > 0, b > 0 \), we can follow these steps:

### Step 1: Identify the general term in the expansion
The general term in the binomial expansion of \( (ax^{1/6} + bx^{-1/3})^9 \) is given by:
\[ T_r = \binom{9}{r} (ax^{1/6})^r (bx^{-1/3})^{9-r} \]
This simplifies to:
\[ T_r = \binom{9}{r} a^r b^{9-r} x^{\frac{r}{6} - \frac{9-r}{3}} \]

### Step 2: Simplify the exponent of \( x \)
The exponent of \( x \) in \( T_r \) is:
\[ \frac{r}{6} - \frac{9-r}{3} = \frac{r}{6} - \frac{27 - 3r}{9} = \frac{3r - 27 + 3r}{18} = \frac{6r - 27}{18} = \frac{2r - 9}{6} \]
We want to find the term independent of \( x \), which means we set the exponent to zero:
\[ 2r - 9 = 0 \implies r = \frac{9}{2} \]
Since \( r \) must be an integer, we consider \( r = 4 \) and \( r = 5 \).

### Step 3: Calculate the terms for \( r = 4 \) and \( r = 5 \)
1. For \( r = 4 \):
\[ T_4 = \binom{9}{4} a^4 b^{5} x^{0} = \binom{9}{4} a^4 b^5 \]
2. For \( r = 5 \):
\[ T_5 = \binom{9}{5} a^5 b^{4} x^{0} = \binom{9}{5} a^5 b^4 \]

### Step 4: Calculate the binomial coefficients
\[ \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]
\[ \binom{9}{5} = \binom{9}{4} = 126 \]

### Step 5: Express the terms
Thus, we have:
\[ T_4 = 126 a^4 b^5 \]
\[ T_5 = 126 a^5 b^4 \]

### Step 6: Find the maximum value
We need to maximize \( T_4 \) and \( T_5 \) under the constraint \( a^2 + b = 2 \).

Using the method of Lagrange multipliers or substituting \( b = 2 - a^2 \) into the expressions for \( T_4 \) and \( T_5 \), we can find the maximum values.

### Step 7: Substitute \( b = 2 - a^2 \) into \( T_4 \)
\[ T_4 = 126 a^4 (2 - a^2)^5 \]

### Step 8: Differentiate and find critical points
To find the maximum, differentiate \( T_4 \) with respect to \( a \) and set it to zero.

### Step 9: Solve for \( a \) and \( b \)
After solving, we find the values of \( a \) and \( b \) that maximize \( T_4 \) or \( T_5 \).

### Step 10: Calculate the maximum value
Finally, substitute the optimal values of \( a \) and \( b \) back into \( T_4 \) or \( T_5 \) to find the maximum value.