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): |
Comment |
| 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
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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:
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| 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 | ||