Let 0 ≤ b ≤ 4.Show that the area included by the two curves y = 1 - | 1 – x | and y = | 2x – b |?

Question

Let 0 ≤ b ≤ 4.Show that the area included by the two curves y = 1 - | 1 – x | and y = | 2x – b |
can not exceed (1/3)

Answer 1

I think it should be |2x + b| instead of |2x - b| because as you have it, there is no way those two curves can have any included area.

You just have to find the value of b which maximizes the area, and it will be b = 2.
So graphs 1-|1-x| and |2x+2| intersect at x = -2/3, -4/3.
The included area is a kite with area = 1/3.

Any other value of b will reduce the included area. In fact, it can be shown that there is no need to specify a range for b since the answer will be the same even if there is no restriction on b.

Answer 2

How far did you get on this by yourself? I'll *help*, but I'm not going to reproduce the entire answer -- that's cheating.

Have you drawn a graph of this to help? Pick an arbitrary value for b (1.5 is a good arbitrary value :-) ) and graph the curves. Can you develop a generalized equation for the area, a function of b?

Answer 3

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