Hey, I posted this question before, but I really need some help.

Thanks...

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A delivery business plans to serve three towns on a stretch of highway. For every trip they make to town A, which we'll place at position x=0, they expect to make two trips to town B (at position x=1) and 3 trips to town C (at position x=4). If they build their delivery centre at position x, then, their daily driving time should be proportional to d(x) = |x| + 2|x - 1| + 3|x - 4|. The owners are thinking of building the center in town C, since it's the biggest but overhead costs would be lower in town B. Where should they build?
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Well, I think I should sketch the function too, but I don't know how...

first, when you make a "trip", one goes there and comes back, so the distance is 2 times ..

On the last question, do the cost function for driving. You will see the cost difference for the initial position, first for x=0, then for x=3

Well, I graphed d(x).

I'm not sure how I am supposed to use it to determine the best position...

Sorry, I don't understand how to do the cost function, since no cost information is given...

To sketch the function d(x) = |x| + 2|x - 1| + 3|x - 4|, you can follow these steps:

1. Identify the key points: Find the x-values where the absolute value terms change. In this case, the absolute value terms change at x = 0, x = 1, and x = 4.

2. Plot the key points: Place a point on the graph for each of the key x-values. For x = 0, d(x) = |0| + 2|0 - 1| + 3|0 - 4| = 7. So plot (0, 7) on the graph. For x = 1, d(x) = |1| + 2|1 - 1| + 3|1 - 4| = 3. So plot (1, 3) on the graph. For x = 4, d(x) = |4| + 2|4 - 1| + 3|4 - 4| = 10. So plot (4, 10) on the graph.

3. Connect the points: Draw a smooth curve that connects the points you plotted. Make sure the curve is increasing or decreasing between the key points.

4. Analyze the graph: Examine the shape of the graph to determine the minimum or maximum point. In this case, the graph is concave up (U-shaped) between x = 1 and x = 4. Therefore, the minimum point should be somewhere between those x-values.

To find the exact coordinates of the minimum point, you can use calculus. Take the derivative of d(x), set it equal to zero, and solve for x.