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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.6b
Chapter 4, Problem 4.5.6b

Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.
b. Find the absolute minimum value of S subject to the given constraint.

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1
First, express the constraint equation xy = 50 in terms of one variable. Solve for y in terms of x: y = 50/x.
Substitute y = 50/x into the objective function S = x + 2y to express S in terms of x alone: S = x + 2(50/x).
Simplify the expression for S: S = x + 100/x.
To find the critical points, take the derivative of S with respect to x. Use the power rule and the derivative of x^(-1) to find dS/dx.
Set the derivative dS/dx equal to zero and solve for x to find the critical points. Check these points to determine the absolute minimum value of S.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Objective Function

An objective function is a mathematical expression that defines a quantity to be maximized or minimized. In this case, S = x + 2y is the objective function, which we aim to minimize while adhering to certain constraints. Understanding how to manipulate and evaluate this function is crucial for finding optimal solutions.
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Constraints

Constraints are conditions that the variables in an optimization problem must satisfy. Here, the constraint xy = 50 restricts the values of x and y, ensuring they remain positive. Recognizing how constraints affect the feasible region is essential for determining the minimum value of the objective function.
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Lagrange Multipliers

Lagrange multipliers are a method used in optimization to find the local maxima and minima of a function subject to equality constraints. This technique involves introducing a new variable (the multiplier) to incorporate the constraint into the optimization process. Applying this method will help in finding the absolute minimum value of S under the given constraint.
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