Question

I don't understand this question

Answer 1

Consider as an example the function
`f(x)=-x^2`. At
`x=-2` and
`x=2`, you have
`f(-2)=f(2)=-4`. The average rate of change of the function over the given interval is


`(f(2)-f(-2))/(2-(-2))=\frac{-4-(-4)}4=0`

so Hunter is correct.

Maggie is also correct, since
`f(x)` has a turning point at the parabola's vertex when
`x=0`.

For a more general situation, you can invoke Rolle's theorem, which states that for a continuous function
`f(x)` over an interval
`[a,b]` with
`f(a)=f(b)` (which is the case for this example) that there is some
`c` in the open interval
`(a,b)` such that
`f'(c)=0`.

Whenever
`f(a)=f(b)`, it's always true that the average rate of change will be
`0`. Provided the function isn't constant, it will always attain an extremum within the given interval, so a turning point must exist.