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.
The graph of the equation looks like this
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 is
can be thought of as so a=1, b=-1, c=0
So 1/2 is the number in the interval [-2, 2] so that the difference of the
number from its square is maximized.
Edwin
.
Find the number in the interval [ - 2, 2 ] so that the difference of the
number from its square is maximized.
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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 = (red), y = x (green) and the difference y = (blue)