Problems

Maximum/Minimum Problems

MathsDirect

We have seen that a maximum or minimum value occurs when the gradient is zero. This gives us a practical method for optimizing values.

For example, if you have a certain amount of wood, what is the largest box you can make.

Or, to look at it the other way; you need to make a 500ml tin. What is the smallest area,( and therefore the least amount of metal,) that the tin can have?

Answering these questions usually involves the following steps

1 Forming an equation to describe the situation.

2 Differentiating the equation.

3 Finding the value that makes the derivative equal zero.

4 Substituting this value into your original equation.

Below are two example problems.

A tin must have a capacity of
What is the smallest surface area that it can have?

First we need to write down the formulas for the volume and surface area

We wish to differentiate, but we have 3 variables, h, r and A.

We can eliminate h, by making it the subject of the Volume equation, then substituting this into the Area equation.


	Substitute h into A and tidy up.
	Differentiate w.r.t. r and say that the result must equal zero.
	Solve the equation that this forms, to find the required radius.
	Substitute this value of r, back into the equation for area.
	and give your final answer.


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