121616949-math.129 - 6.1 Optimization 115 SummarySteps to...

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6.1 Optimization 115 Summary—Steps to solve an optimization problem. 1. Decide what the variables are and what the constants are, draw a diagram if appropriate, understand clearly what it is that is to be maximized or minimized. 2. Write a formula for the function for which you wish to find the maximum or minimum. 3. Express that formula in terms of only one variable, that is, in the form f ( x ). 4. Set f ( x ) = 0 and solve. Check all critical values and endpoints to determine the extreme value. Exercises 1. Let f ( x ) = 1 + 4 x - x 2 for x 3 ( x + 5) / 2 for x > 3 Find the maximum value and minimum values of f ( x ) for x in [0 , 4]. Graph f ( x ) to check your answers. 2. Find the dimensions of the rectangle of largest area having fixed perimeter 100. 3. Find the dimensions of the rectangle of largest area having fixed perimeter P . 4. A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. 5.
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