Question

If \( x + y + z = 45 \) and the maximum value of \( x^4 y^6 z^5 \) exists at \( x = \alpha,\, y = \beta,\, z = \gamma \), then the value of \[ \frac{\alpha + \beta}{\gamma} = ? \]

• A. 2
• B. 1
• C. 3
• D. 4

Hint:

Use Lagrange multipliers to optimize a product under a linear constraint. Equate partial ratios and substitute.

Answer 1

The Correct Option is A

Given \( x+y+z=45 \) and we need to find the maximum value of \( x^4y^6z^5 \). This is a case of using the method of Lagrange multipliers or the AM-GM inequality for optimization.

Using AM-GM inequality, the product \( x^4y^6z^5 \) is maximized when the terms are proportional to the powers in the product. This implies \( \frac{x}{4}=\frac{y}{6}=\frac{z}{5} \).

Let \( x=4k \), \( y=6k \), and \( z=5k \). Then:

\[ x+y+z = 4k+6k+5k = 15k = 45 \]

Solving for \( k \):

\( k = \frac{45}{15} = 3 \)

Substitute back to find \( x, y, z \):

\( x = 4 \times 3 = 12 \)

\( y = 6 \times 3 = 18 \)

\( z = 5 \times 3 = 15 \)

Thus, \( \alpha = 12 \), \( \beta = 18 \), \( \gamma = 15 \) and:

\[ \frac{\alpha + \beta}{\gamma} = \frac{12 + 18}{15} = \frac{30}{15} = 2 \]

The correct answer is 2.