### Calculus meets Functions

Many ways problem

## Problem

Each column and row heading in the following table is a property that a function may or may not have. A function can appear in a cell if it has the properties in the corresponding row and column.

We have omitted some headings, and some entries in cells. Can you complete the table?

You might find it helpful to draw some sketches. You could use graph-sketching software such as Desmos to help you, but try to do the sketching by hand before reaching for a computer or calculator!

Make sure that you can explain why each function has the desired properties.

The curve is $\ldots$creasing for $x>1$ Has a local $\ldots$imum with
$y$-coordinate $1$
Has a stationary point at $(1,1)$ $y=x^3+3x^2-9x+6$ $y=3x^4-4x^3+2$
$y=2x^3+9x^2+1$ $y=1-\dfrac{1}{4}x^4-x^3$
Has an $\,\,\ldots \,\,$ number of stationary points $y=x^5-x^3+5$
• Can you complete the table using a different function in every cell?

• On the other hand, can you complete the table with a smaller number of functions?

• Did you have any choice about the column and row headings?