Suppose that we have following interval (−5,2),we should find such a, which takes all possible values from this interval,creates following inequality systems 5 + a − | 2 y | ≥ 0 | x | ≤ | a − 2 | 2

Question

Suppose that we have following interval (−5,2),we should find such a, which takes all possible values from this interval,creates following inequality systems  5  +  a  −  |  2  y  |  ≥  0  |  x  |  ≤                |      a      −      2          |        2

Esteban Johnson · Accepted Answer

Let&#039;s take an example, it may clarify things a bit for you. Suppose   a  =  0. Then   |  y  |  ≤  5  /  2, and   |  x  |  ≤  1  /  2, so the figure we are talking about is the rectangle bounded by the horizontal lines   y  =  5  /  2 and   y  =  −  5  /  2, and the vertical lines   x  =  −  1  /  2 and   x  =  1  /  2. This rectangle has sides 5 and 1, and area 5.Now try it for some other value of   a, like   a  =  1 or   a  =  −  1, and see what you get.Then try to get a formula that works for all values of   a.We&#039;re told   −  5  &amp;lt;  a  &amp;lt;  2, so   a  −  2 is always negative, so   |  x  |  ≤  (  2  −  a  )  /  2, so            a      −      2        2    ≤  x  ≤            2      −      a        2  Also,   5  +  a is always positive, so   |  y  |  ≤  (  5  +  a  )  /  2, so  −            5      +      a        2    ≤  y  ≤            5      +      a        2  So the area in question is a rectangle with sides   2  −  a and   5  +  a, hence, areaMaximizing the quadratic is an exercise left to the reader.

Suppose that we have following interval (−5,2),we should find such a, which takes all possible value

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