Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Differentiation - Applications of Calculus.
Maximum-minimum - Main page.


There are many types of questions asking you to determine a maximum and/or minimum value.
Many of these either give the equation of a function or ask you to develop such an equation.

The pages hyperlinked in the tables below reference problems based on ordinary functions.
Others questions incorporate:

trigonometric functions;
exponential functions; or
logarithmic functions

(and these can be located using the hyperlinks just provided).

For all max-min questions, there is a common set of steps to be followed with only one possible exception. These steps are summarised as follows:

When given a function f(x):

1. Determine the 1st derivative f '(x).  
2. If it can be done, determine the 2nd derivative immediately.  
3. The 2nd derivative often allows you to interpret the concavity at this stage in terms of whether the curve has a maximum or minimum. If so, do this interpretation


It is often possible to draw a conclusion from the 2nd derivative about a function having maximum or minimum value before solving the first derivative. For example -x2 is always negative and t + 2 is always positive when t is time. Just do the algebra.
4. Solve the 1st derivative = 0. This step reveals the values at which the gradient of the function equals zero - and therefore possible values for x at which there are maximum or minimum values or horizontal points of inflection.
5. If it were not possible to interpret the concavity in step 3, use the values calculated in step 4 to substitute into the 2nd derivative equation and determine the concavity. Remember:
  • +ve concavity shows a minimum value;
  • -ve concavity indicates a maximum value.
6. If required, find the horizontal points of inflection. As the 1st and 2nd derivatives equal zero, you need to use a table of three x values to show there is a change in the sign of the concavity.

These steps are followed in all of the exercises which are associated with this page - see the Differentiation - Main page link below.


Problem type 1: given the equation of a curve:

Equations of curves - Test Yourself 1.
Equations of curves - Test Yourself 1 - Solutions.
Equations of curves - Test Yourself 2.
Equations of curves - Test Yourself 2 - Solutions.


Problem type 2: Applied maximum/minimum questions:

Type Nature of questions Test Yourself 1 Test Yourself 2
Type 1: Basic numbers. Test 1      
Type 2: 2D shapes - eg paddocks, geometrical shapes, distances between curves, etc Test 1 Solutions 1 Test 2 Solutions 2
Type 3: Simple 3D shapes - eg a cylinder, a trough, etc. Test 1 Solutions 1 Test 2 Solutions 2

Type 4:

Advanced 3D shapes with one shape inside another. Test 1 Solutions 1    
Type 5: Uses rates - eg Cost per unit, metres/sec, etc Test 1 Solutions 1 Test 2 Solutions 2
Type 6: Miscellaneous Test 1 Solutions 1