4.1 Critical Values and Optimization

When do critical points occur?

• When f'(x)=0

• When f(x)=0

• When f''(x)=0

• When f'(x)=0 or undefined

Answer explanation

We need to check where the derivative equals 0 (horizontal tangents) as well as when the derivative is undefined for critical points.

If a function is a cubic function, its derivative is a...

• quartic function

• quadratic function

• linear function

• constant function

Answer explanation

When you take the derivative of a function, you lose a degree. A cubic (third-degree polynomial) becomes a quadratic (second-degree polynomial).

In optimization problems, you are really finding...

• x-intercepts

• extrema

• points of inflection

• volume formulas

In optimization problems it is important that the function you are maximizing or minimizing has...

• one variable

• two variables

• lots of exponents

• pi

Answer explanation

Make sure the function you are taking the derivative only has one variable in it. If it doesn't, use your constraint to rewrite the function in terms of one variable.

You want to make a box to contain dirt and your pet earthworm. Using a 7 in by 10 in rectangle of cardboard, you cut congruent squares from the corners and fold up the sides. Choose the equation you would use in order to do Calculus to find the maximum volume of dirt (including worm) the box can hold.

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem. 

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

• 100ft x 200ft

• 102ft x 196 ft

• 50 ft x 300 ft

• 50 ft x 175 ft

In order for the Extreme Value theorem to apply, which of these must be true. Select all that apply.

• It must be discontinuous

• It must be closed

• It must be open

• It must be continuous

To find absolute extrema, you must include...

• x-intercepts

• y-intercepts

• points of inflection

• endpoints

Answer explanation

To find absolute maxima and minima, find your critical points (relative extrema) and test them against the endpoints by finding their values. Plug endpoints and critical points into the original function. The highest point is the absolute maximum. The lowest point is the absolute minimum.