### There are 17 results

Broad Topics >

Sequences, Functions and Graphs > Maximise/minimise/optimise

##### Age 16 to 18 Challenge Level:

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?

##### Age 16 to 18 Challenge Level:

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.

##### Age 16 to 18 Challenge Level:

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the
parameter t varies.

##### Age 16 to 18 Challenge Level:

Can you massage the parameters of these curves to make them match as closely as possible?

##### Age 16 to 18 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

##### Age 16 to 18 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of
1/u + 1/v given that u + v = 5 by different methods.

##### Age 16 to 18 Challenge Level:

A point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
vertices?

##### Age 14 to 16 Challenge Level:

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

##### Age 16 to 18 Challenge Level:

Four rods are hinged at their ends to form a quadrilateral. How can you maximise its area?

##### Age 14 to 16 Challenge Level:

There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Make and prove a conjecture about the cyclic quadrilateral
inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

##### Age 16 to 18 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
corner.

##### Age 14 to 16 Challenge Level:

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

##### Age 14 to 16 Challenge Level:

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

##### Age 16 to 18 Challenge Level:

Can you work out how to produce the right amount of chemical in a
temperature-dependent reaction?