SOLUTION: Find the number in the interval [ - 2, 2 ] so that the difference of the number from its square is maximized.

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Question 1181193: Find the number in the interval [ - 2, 2 ] so that the difference of the
number from its square is maximized.
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20077) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be the number in the interval [-2, 2] so that the difference of the
number from its square is maximized. 

the difference of the number x from its square x2 is x2 - x.

Let y equal to this difference x2 - x.
 
y%22%22=%22%22x%5E2-x

The graph of the equation looks like this

graph%28400%2C400%2C-3%2C3%2C-3%2C3%2Cx%5E2-x%29

The value of y (the difference) is minimized at the bottom of
that graph (at the bottom point), which is the vertex.

The formula for the x-coordinate of the vertex of y=ax%5E2%2Bbx%2Bc is 

     -b%2F%282%2Aa%29

y=x%5E2-x can be thought of as y=1a%5E2-1x%2B0 so a=1, b=-1, c=0

     -b%2F%282a%29
     -%28-1%29%2F%282%281%29%29
     1%2F2

So 1/2 is the number in the interval [-2, 2] so that the difference of the
number from its square is maximized. 

Edwin

Answer by ikleyn(53751) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the number in the interval [ - 2, 2 ] so that the difference of the
number from its square is maximized.
~~~~~~~~~~~~~~~~~~~~


            The solution given by  Edwin in his post is  ONLY  PART  of the FULL  solution,  and being the part, only,
            IT  DOES  NOT  GIVE  the full solution.

            His solution for the local minimum/maximum should be supplemented by the end-behavior analysis
            of the function  f(x) = x^2 - x. 


This end-behavior analysis gives the values


    (a)  at x = -2,  f(-2) = (-2)^2 - (-2) = 4 + 2 = 6;


    (a)  at x =  2,  f(2)  =   2^2  -   2  = 4 - 2 = 2.


So, the answer is:  in the interval [-2,2],  the number which provides the maximum of the function f(x) = x^2 - x, is the value of x= -2.


As you see,  the correct answer is totally different from that by  Edwin.


                            For better understanding,  see the plot below. 


              


     Plot y = x%5E2 (red),  y = x (green)  and  the difference y = x%5E2+-+x (blue)