Question

If \( x \) and \( y \) are two positive integers such that \( x + y = 24 \) and \( x^3 y^5 \) is maximum, then \( x^2 + y^2 \) is:

• A. 288
• B. 296
• C. 306
• D. 320 \bigskip

Hint:

To maximize a product of terms, convert it to a function of one variable using the constraint and apply differentiation.

Answer 1

The Correct Option is C

We are given that \( x + y = 24 \) and \( x^3 y^5 \) is maximum. To maximize \( x^3 y^5 \), we use the method of Lagrange multipliers or simply apply the optimization conditions for a product. Let \( y = 24 - x \). Then, the function to be maximized is: \[ f(x) = x^3 (24 - x)^5. \] To maximize this, we take the derivative of \( f(x) \) and set it to zero: \[ f'(x) = 3x^2(24 - x)^5 - 5x^3(24 - x)^4. \] Simplifying: \[ f'(x) = x^2 (24 - x)^4 \left( 3(24 - x) - 5x \right). \] \[ f'(x) = x^2 (24 - x)^4 (72 - 8x). \] Setting \( f'(x) = 0 \), we get: \[ 72 - 8x = 0 \quad \Rightarrow \quad x = 9. \] Substituting \( x = 9 \) into \( x + y = 24 \), we get \( y = 15 \). Thus, \( x = 9 \) and \( y = 15 \). Now, calculate \( x^2 + y^2 \): \[ x^2 + y^2 = 9^2 + 15^2 = 81 + 225 = 306. \] \bigskip