Also, for which [imath]x[/imath] are those minimums achieved?Whats the minimum of [imath]|x| + |x+a|[/imath]? How about [imath]|x+a|+|x+b|[/imath] ?
The minimum of |x| is a number, not an expression; and it occurs at a specific value of x. It sounds like you have the wrong image of what is being asked.Thanks! The minimum for |x| = -x and the minimum of |x+a| will be -(x+a), right?
Thank you for your illuminating response (and for affirming that I was on the right track).There's an easier way to approach it, without having to graph too many of these (by hand or by technology) -- though in fact I started out by doing part of what you did to visualize what's happening.
Yes, I was a bit concerned that I was doing a lot of the 'donkey work' for the OP but I was intrigued by the problem and, once I'd started pursuing your suggestion, I was keen to know that I was implementing it correctly and, having done all that graph work (and figured out a formula), it was too difficult to resist seeking confirmation that I'd got it right.(But let's not give away quite so much to the OP, shall we?)
Indeed, that's similar to how I would illustrate arriving at the formula for calculating the sum of the first n natural numbers (another potential hint for the OP ).My trick for getting a quick answer is not to start with |x| + |x+1|, but to break the sum up into |x| + |x+55|, |x+1| + |x+54|, and so on. This sort of symmetrical approach helps with a lot of problems.
And that's perfectly true for "only odd final numbers" but not for even final numbers, of course, which is what does make the problem that little bit "tricky" and why I would expect the OP to have to show, algebraically, how a final formula can be arrived at given all the 'raw data' that has been provided.To see the pattern more quickly, you can consider only odd final numbers...
I do get the same result, but in a simpler way. (It's just a perfect square.)
It's definitely worth putting some thought into; as you can see, there are several ways to think about it, and each may teach you something a little different. The first thing is to develop a sense of what functions like this are: continuous piecewise linear functions. (I suppose that's the "conceptual" part!) Then once you've got a good idea of that, you have to think about how to handle a sum of terms that are too many to write down all of; this brings you into the area of sequences and series, and looking for patterns. (That would be the "challenging" part.) So you've described it well.The solution seems to be both conceptual and super challenging. I need to spend time to understand the solution and grasp the concept. Thank you.
Thanks, that makes sense!It's definitely worth putting some thought into; as you can see, there are several ways to think about it, and each may teach you something a little different. The first thing is to develop a sense of what functions like this are: continuous piecewise linear functions. (I suppose that's the "conceptual" part!) Then once you've got a good idea of that, you have to think about how to handle a sum of terms that are too many to write down all of; this brings you into the area of sequences and series, and looking for patterns. (That would be the "challenging" part.) So you've described it well.
Feel free to show us any further ideas you have; there's definitely more to discuss when you're ready for it.
That sounds awfully like a 'sign off', @gamaz321!Thanks, that makes sense!