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### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Cubics

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

### Patterns of Inflection

Find the relationship between the locations of points of inflection, maxima and minima of functions.

# Cubic Spin

##### Stage: 5 Challenge Level:

If the graph of the cubic polynomial has rotational symmetry then a maximum point must be rotated to become a minimum and vice versa so the center of rotation will be the midpoint of the line joining the maximum and minimum points. If there are no maximum and minimum points then consider the point of inflexion.

The graph of a function has rotational symmetry about the origin if and only if $f(-x) = -f(x)$. You can do this question without calculus if you can find the transformation of coordinates that removes the quadratic term from the polynomial equation.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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