Maximum and minimum problems

By the end you can answer "find the maximum volume" (or the least area, or the greatest product) in a word problem, and show that it really is a maximum.

A typical question: Show that the volume of the box is $V = 144x - 48x^2 + 4x^3$ . Find the maximum volume, and show that it is a maximum.

Giveaway words: greatest, least, maximum, minimum, largest possible. There's often a show that first.

It's the stationary-points method again, with a word problem wrapped round it.

The same four steps every time:

Formula: write the thing they want (volume, area…) using one letter. Use the fixed fact they give you (a fixed perimeter, a fixed volume) to get rid of any second letter.
Solve: differentiate, set it equal to 0 and solve. Throw out impossible values, like zero or negative lengths.
Prove: differentiate again. Negative means maximum, positive means minimum. Say so in words.
Answer: put the value back in to get the maximum volume (or area, or cost) they asked for, with units.

A rectangle has a perimeter of 20 m. Find its greatest possible area.

Positive numbers $x$ and $y$ satisfy $x + y = 12$ . Find the maximum value of $P = x^2y$ .

A rectangular pen is built against a long wall, using 40 m of fencing for the other three sides. The two sides perpendicular to the wall are each $x$ m and the side parallel to it is $y$ m.

A student finds the maximum area. Which line contains the first error?

Maximum and minimum problems

What they ask

The method