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.
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.