Calculus  Differentiation  Applications of Calculus.
Maximumminimum  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 maxmin 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): 
Comment 
1. Determine the 1^{st} derivative f '(x).  
2. If it can be done, determine the 2^{nd} 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 2^{nd} derivative about a function having maximum or minimum value before solving the first derivative. For example x^{2} is always negative and t + 2 is always positive when t is time. Just do the algebra. 
4. Solve the 1^{st} 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 2^{nd} derivative equation and determine the concavity.  Remember:

6. If required, find the horizontal points of inflection.  As the 1^{st} and 2^{nd} 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 