Solve The Problem Maximize Qxy² Where X And Y Are Positive

Solve The Problemmaximize Qxy2 Where X And Y Are Posit

Analyze the problem of maximizing the function Q = xy² where x and y are positive numbers, under the constraint x + y² = 10.

Paper For Above instruction

In the realm of optimization problems in calculus, maximizing functions subject to constraints is a fundamental task. A common approach involves using methods such as substitution or Lagrange multipliers. Here, we are asked to maximize the function Q = xy², with the restriction that x + y² = 10, and both x and y are positive real numbers.

To solve this, we begin by expressing the constraint explicitly as a function of one variable, and then substitute into the objective function. The constraint gives us x = 10 - y². Since x and y are positive, this implies that y is in the interval (0, √10). With x expressed as 10 - y², we substitute into Q:

\[Q(y) = (10 - y^2) y^2 = 10 y^2 - y^4.\]

Next, to find the maximum of Q as a function of y, we differentiate with respect to y and set the derivative equal to zero:

\[\frac{dQ}{dy} = 20 y - 4 y^3 = 0.\]

Factor out common terms:

\[4 y (5 - y^2) = 0.\]

The solutions are y = 0 or y² = 5. Since y > 0, y = √5. Corresponding to y = √5, x = 10 - (√5)^2 = 10 - 5 = 5. The point y = 0 is not considered since y > 0 and it results in Q=0, which cannot be a maximum in this context.

Now, evaluate Q at y = √5:

\[Q = xy^2 = 5 \times 5 = 25.\]

Considering the endpoints, as y approaches 0, Q approaches 0, and at y = √10, the value of Q is less than at y = √5, confirming that the maximum occurs at y = √5.

Therefore, the maximum value of Q is 25, achieved when y = √5 and x = 5. The corresponding y value is approximately 2.236, giving an x of 5, indicating the optimal solution under the given constraints.

References

  • Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
  • Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Brooks Cole.
  • Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendental (10th ed.). Wiley.
  • Thomas, G. B., & Finney, R. L. (2002). Calculus and Analytic Geometry (9th ed.). Pearson.
  • Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
  • Weisstein, E. W. "Optimization." From Wolfram MathWorld. https://mathworld.wolfram.com/Optimization.html
  • Fletcher, R. (2013). Practical Methods of Optimization. Wiley.
  • Bulkas, A. (2020). Optimization Problems in Calculus. Journal of Mathematics Education.
  • Larson, R. & Edwards, B. (2017). Calculus. Cengage Learning.
  • Epstein, M. (2009). Calculus for Business, Economics, and the Social and Life Sciences. McGraw-Hill.

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