Turning points

This problem challenges you to sketch curves with different properties.

This problem challenges you to find cubic equations which satisfy different conditions.

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

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.

Consider these analogies for helping to understand key concepts in calculus.

Can you sketch these difficult curves, which have uses in mathematical modelling?

A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.

What is the quickest route across a ploughed field when your speed around the edge is greater?

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Find the maximum value of n to the power 1/n and prove that it is a maximum.