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Finding the max/min of a curve
A useful application of calculus is finding maximum or minimum value(s) of a function.
In this graph of the function there is a local maximum (at
) and local minimum (at
).
The blue horizontal line shows that the gradient at these points is zero i.e.
Using differentiation:
To find the max/min points make
There are 2 possible solutions, or
How can we tell which solution is the max or min?
Take the second derivative (i.e. differentiate to get
).
To test point :
If ,
is a minimum
If ,
is a maximum
note that if ,
is a point of inflection
In our example:
When ,
to show there is a maximum at
When ,
to show there is a minimum at
To see this working explained more fully see this Khan Academy video.
Further information
- Khan Academy explains how a study of the gradient on various points of a curve will show places where a relative maximum or minimum exist.
- Desmos is a free online calculator which can be used to draw graphs to show significant features like local maximum and minimum.