The cost of manufacturing a container for frozen orange...

The cost of manufacturing a container for frozen orange juice is $C(h)=h^{2}-7.4 h+13.7$ cents, where $h$ is the height of the container in inches. Your boss claims that the containers will be cheapest to make if they are 4 inches tall. Use Theorem 3.3 to quickly show that he is wrong.

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Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form ax² + bx + c, where the constant 'a' determines the direction of the parabola's opening (upward if a > 0, downward if a < 0). The graph of a quadratic function is a parabola, which is a crucial concept for understanding optimization in many real-world applications, such as cost analysis.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its minimum or maximum value. For quadratic functions that open upward, like the one given (since the coefficient of h² is positive), the vertex represents the minimum point. The x-coordinate of the vertex can be determined using the formula -b/(2a), making it a quick method for finding the optimal value in optimization problems.

Optimization in Quadratic Functions

Optimization in the context of quadratic functions involves finding the vertex to determine the minimum or maximum value of the function. Through techniques such as applying the vertex formula or completing the square, one can efficiently identify the optimal parameter value (in this case, the height) that will minimize the cost associated with manufacturing.

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