Section 6: Optimization

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 5.6 Optimization 529 Version: Fall 2007 5.6 Optimization In this section we will explore the science of optimization. Suppose that you are trying to find a pair of numbers with a fixed sum so that the product of the two numbers is a maximum. This is an example of an optimization problem. However, optimization is not limited to finding a maximum. For example, consider the manufacturer who would like to minimize his costs based on certain criteria. This is another example of an optimization problem. As you can see, optimization can encompass finding either a maximum or a minimum. Optimization can be applied to a broad family of different functions. However, in this section, we will concentrate on finding the maximums and minimums of quadratic functions. There is a large body of real-life applications that can be modeled by qua- dratic functions, so we will find that this is an excellent entry point into the study of optimization. Finding the Maximum or Minimum of a Quadratic Function Consider the quadratic function f ( x ) = x 2 + 4 x + 2 . Let’s complete the square to place this quadratic function in vertex form. First, factor out a minus sign. f ( x ) = x 2 4 x 2 Take half of the coefficient of x and square, as in [(1 / 2)( 4)] 2 = 4 . Add and subtract this amount to keep the equation balanced. f ( x ) = x 2 4 x + 4 4 2 Factor the perfect square trinomial, combine the constants at the end, and then redis- tribute the minus sign to place the quadratic function in vertex form. f ( x ) = ( x 2) 2 6 f ( x ) = ( x 2) 2 + 6 This is a parabola that opens downward, has been shifted 2 units to the right and 6 units upward. This places the vertex of the parabola at (2 , 6) , as shown in Figure 1 . Note that the maximum function value ( y -value) occurs at the vertex of the parabola. A mathematician would say that the function “attains a maximum value of 6 at x equals 2.” Note that 6 is greater than or equal to any other y -value (function value) that occurs on the parabola. This gives rise to the following definition. Copyrighted material. See: 1
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530 Chapter 5 Quadratic Functions Version: Fall 2007 x 10 y 10 (2 , 6) Figure 1. The maximum value of the function, 6, occurs at the vertex of the parabola, (2 , 6) . Definition 1. Let c be in the domain of f . The function f is said to achieve a maximum at x = c if f ( c ) f ( x ) for all x in the domain of f . Next, let’s look at a quadratic function that attains a minimum on its domain. l⚏ Example 2. Find the minimum value of the quadratic function defined by the equation f ( x ) = 2 x 2 + 12 x + 12 . Factor out a 2. f ( x ) = 2 x 2 + 6 x + 6 (3) Take half of the coefficient of x and square, as in [(1 / 2)(6)] 2 = 9 . Add and subtract this amount to keep the equation balanced. f ( x ) = 2 x 2 + 6 x + 9 9 + 6 Factor the trinomial and combine the constants, and then redistribute the 2 in the next step.
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