Finding an Optimum

Learn verifying an optimum, interpreting the optimal value.

Lesson 2 of 32 parts Practice Flashcards

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Verifying an Optimum

Objective: Explain verifying an optimum, apply the stated method, and check its conditions.

Solve A′(x)=0 for interior candidates, then verify their nature using sign changes, second derivatives, or global comparisons. For a fixed-perimeter rectangle, the area quadratic is concave down, so its feasible vertex is the unique maximum. A zero derivative is a candidate, not an automatic maximum.

Formulas

A′=0 plus a maximum test

Worked examples

Try itA rectangle has perimeter 28 cm. Find the width maximizing its area.Show answer

Step 1: A(x)=x(14−x). Step 2: A′(x)=14−2x=0 gives x=7. Step 3: A″=−2<0; the maximum-area rectangle is a square.

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Verifying an Optimum

Objective: Explain verifying an optimum, apply the stated method, and check its conditions.

Formulas

A′=0 plus a maximum test

Worked examples

Try itA rectangle has perimeter 28 cm. Find the width maximizing its area.Show answer

Step 1: A(x)=x(14−x). Step 2: A′(x)=14−2x=0 gives x=7. Step 3: A″=−2<0; the maximum-area rectangle is a square.

Interpreting the Optimal Value

Objective: Explain interpreting the optimal value, apply the stated method, and check its conditions.

An optimization answer can request the input design, the objective value, or both. After finding the best side lengths, substitute them into the original area model. Report area in square units and distinguish it from an optimal length in linear units. Answer the requested quantity with its correct units.

Formulas

Amax=(P/4)² for a fixed-perimeter rectangle

Worked examples

Try itFind the largest area of a rectangle with perimeter 28 cm.Show answer

Step 1: The largest area occurs for equal side lengths. Step 2: Each side is (28)/4=7 cm. Step 3: The maximum area is 49 cm².

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