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I don't understand this question

I don't understand this question-example-1
User ChargerIIC
by
7.8k points

1 Answer

1 vote
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.
User Peter Schorn
by
7.7k points

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