Extreme Value Theorem (College Board AP® Calculus AB)

When using the extreme value theorem on the exam:

• Be sure to justify that the theorem is valid

  • I.e. that the function is continuous on

• Remember that if a function is differentiable on an interval

  • then it is also continuous on that interval

A social sciences researcher is using a function to model the total mass of all the garden gnomes appearing on lawns in a particular neighborhood at time . The function is twice-differentiable, with measured in kilograms and measured in days.

The table below gives selected values of over the time interval .

Justify why there must be at least one time, , for , at which , the rate of change of the mass, equals 0 kilograms per day.

Answer:

You need to prove that the derivative has a particular value at an unspecified point

At first this might seem like a job for the mean value theorem

But there are no two points in the table between which the average rate of change is zero

Instead, the stated result may be proved by using the extreme value theorem

Start by showing that is continuous; then the extreme value theorem will be valid

Remember that differentiability implies continuity

differentiable continuous on

The extreme value theorem says that will have a maximum value on

Show that the maximum value does not occur at either of the endpoints of the interval

and

Because is differentiable on , this means the maximum is a local maximum point somewhere in that open interval, at which point is equal to zero

is twice-differentiable, which means is differentiable, which means is continuous on

By the extreme value theorem, has a maximum on

That maximum does not occur at or , therefore has a maximum on the interval

Because is differentiable, must equal 0 at this maximum