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

We are to maximize \( x^4 y^6 z^5 \) subject to the constraint \( x + y + z = 45 \). This is a standard constrained optimization problem using Lagrange multipliers. Let \( f(x, y, z) = x^4 y^6 z^5 \), and \( g(x, y, z) = x + y + z - 45 = 0 \) Using Lagrange multipliers, we solve: \[ \nabla f = \lambda \nabla g \Rightarrow \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = \lambda (1, 1, 1) \] \[ \Rightarrow \frac{4}{x} = \frac{6}{y} = \frac{5}{z} \Rightarrow x : y : z = \frac{1}{4} : \frac{1}{6} : \frac{1}{5} \Rightarrow 15 : 10 : 12 \] Let \( x = 15k, y = 10k, z = 12k \). Then: \[ x + y + z = 15k + 10k + 12k = 37k = 45 \Rightarrow k = \frac{45}{37} \] Now, \[ \frac{\alpha + \beta}{\gamma} = \frac{x + y}{z} = \frac{15k + 10k}{12k} = \frac{25k}{12k} = \frac{25}{12} \approx 2.08 \Rightarrow \boxed{2} \]