Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Can You Find... Cubic Curves

**This is an Underground Mathematics resource.**

*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*

*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*
## You may also like

### Powerful Quadratics

### Discriminating

### Factorisable Quadratics

Or search by topic

Age 16 to 18

Challenge Level

- Problem
- Student Solutions

Can you find a cubic curve that...

(a) ... has no stationary points?

(b) ... has two stationary points: one when $x=2$ and one when $x=5$?

(c) ... has a local minimum when $x=-1$?

(d) ... has a local minimum when $x=-2$ and a local maximum when $x=4$?

It would be a good idea to try and *sketch* some of the cubics first before trying to form an equation.

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.