Find the number in the interval [-1,1] such that the difference of the number minus its square is
a.) A maximum
b.) a minimum
Support your answers graphically.
Hello,
to find an extremum (maximum or minimum) you have to examine the borders of the interval and the drivative of the function.
Let x be the number you are looking for.
Then you get the difference by:
$\displaystyle d(x)=x-x^2$
$\displaystyle d'(x)=1-2x$. If d'(x) = 0 then x = 1/2 and the difference is d(1/2) = 1/4. By the 2nd drivative (it is negative) you know that this must be the maxium.
Now examine the borders of the given interval:
d(1) = 0
d(-1) = -2
-2 is smaller than 1/4 so this must be the minimum.
EB