# Calculus/Differentiation/Applications of Derivatives/Solutions

## Relative ExtremaEdit

Find the relative maximum(s) and minimum(s), if any, of the following functions.

There are no roots of the derivative. The derivative fails to exist when x=-1 , but the function also fails to exists at that point, so it is not an extremum. Thus,

**the function has no relative extrema.**

There are no roots of the derivative. The derivative fails to exist at . .

**The point is a minimum**since is nonnegative because of the even numerator in the exponent.

**The function has no relative maximum.**

Since the second derivative is positive, ** corresponds to a relative minimum.**

**There is no relative maximum.**

Since the second derivative of at is positive, ** corresponds to a relative mimimum.**

**corresponds to a relative maximum.**

**corresponds to a relative minimum. There is no relative maximum.**

Since is positive, ** corresponds to a relative minimum.**

**corresponds to a relative maximum.**

## Range of FunctionEdit

Since is negative, corresponds to a relative maximum.

For , is positive, which means that the function is increasing. Coming from very negative -values, increases from a very negative value to reach a relative maximum of at .

For , is negative, which means that the function is decreasing.

Since is positive, corresponds to a relative minimum.

Between the function decreases from to , then jumps to and decreases until it reaches a relative minimum of at .

For , is positive, so the function increases from a minimum of .

## Absolute ExtremaEdit

Determine the absolute maximum and minimum of the following functions on the given domain

Check the endpoint:

**Maximum at ; minimum at**

**Maximum at ; minimum at**

## Determine Intervals of ChangeEdit

Find the intervals where the following functions are increasing or decreasing

is the equation of a line with negative slope, so is positive for and negative for .

**increasing on and decreasing on**.

is the equation of a bowl-shaped parabola that crosses the -axis at and , so is negative for and positive elsewhere.

**decreasing on and increasing elsewhere.**

is the equation of a hill-shaped parabola that crosses the -axis at and , so is positive for and negative elsewhere.

**increasing on and decreasing elsewhere.**

**increasing on and decreasing elsewhere**.

is negative on and positive elsewhere.

**decreasing on and increasing elsewhere.**

**decreasing on and increasing elsewhere.**

## Determine Intervals of ConcavityEdit

Find the intervals where the following functions are concave up or concave down

**concave down everywhere.**

When , is negative, and when , is positive.

**concave down on and concave up on .**

is positive when and negative when .

**concave up on and concave down on .**

**concave up on and concave down on .**

is positive when and negative when .

**concave down on and concave up on .**

is positive when and negative when .

**concave down on and concave up on .**

## Word ProblemsEdit

After 4 seconds, the rate of change in position with respect to time is

The distance between the bikes is given by

Let represent the elapsed time in hours. We want when . Apply the chain rule to :

**13 mph**.

We are constricted to have a can with a volume of , so we use this fact to relate the radius and the height:

The surface area of the side is

and the cost of the side is

The surface area of the top and bottom (which is also the cost) is

The total cost is given by

We want to minimize , so take the derivative:

Find the critical points:

Check the second derivative to see if this point corresponds to a maximum or minimum:

Since the second derivative is positive, the critical point corresponds to a minimum. Thus, the minimum cost is