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Maxima and Minima
By Frank on May 4, 1998:

I'm a year 12 student and I'm having some trouble understanding maximum and minimum values and how they're derived from differentiation. My book is terrible, it doesn't explain anything!

thanks..

Frank.

By Jo on May 4, 1998:

Hopefully the following will help.

y=f(x) describes a curve in the x-y plane, for example, y=x2 has the following graph:
parabola

You can see that the slope or gradient of the graph varies as you move along it.

You can find the gradient at a particular point on the graph by differentiating its equation.

For example, if y=x2 then differentiating gives dy/dx=2x, so the gradient at x=1 is 2×1=2.


Maximum and minimum values:
These are points on the graph where dy/dx=0, ie the tangent to the graph is parallel to the
x-axis. You can see that at these points the graph appears to ``turn around'', e.g. y=x2 has a turning point at x=0. You can tell whether you have a maximum or a minimum by drawing a graph: a minimum looks like the y=x2 graph and a maximum is the other way up.


Summary: Find maximum and minimum points as follows:

  1. differentiate equation of graph
  2. set this equal to zero.
  3. solve for x to give the position of the turning points.
  4. draw a graph to see whether each turning point is a maximum or a minimum.


If you are interested, an alternative method for distinguishing between maxima and minima is to look at values either side of the turning point. If the values of the function on each side of the equation are less than the value at the turning point then you have a maximum. If they are more then you have minimum. There is also a more complicated method where you differentiate for a second time, but I suspect that will come later in your course.

Jo