Thread: Maximum and minimum points of a curve

Hey,

I've just been doing some differentiation questions. I understand that the points where dy/dx = 0 wil be turning points on the graph, but I don't understand why they are called "minimum" and "maximum" points.

For example, in the graph of the equation y = 2x^3 + 3x^2 -36x+1
the maximum turning point is (-3, 82), but the line reaches higher points in the y-axis than this further along the x-axis. So why is it called a maximum point?

My guess is that it just means maximum turning point. Is this correct? If so, are there no further turning points anywhere along the line?

Originally Posted by russell_c_cook

Hey,

I've just been doing some differentiation questions. I understand that the points where dy/dx = 0 wil be turning points on the graph, but I don't understand why they are called "minimum" and "maximum" points.

For example, in the graph of the equation y = 2x^3 + 3x^2 -36x+1
the maximum turning point is (-3, 82), but the line reaches higher points in the y-axis than this further along the x-axis. So why is it called a maximum point?

My guess is that it just means maximum turning point. Is this correct? If so, are there no further turning points anywhere along the line?

They are not called maximum or mimimum points unless they really are maximums or minimums.

They are called critical points. There are three types of critical points 1) relative maxima (aka local maxima) 2) relative minima (aka local minima) and 3) inflection points.

A relative maximum is a point at which the function is greater than or equal to its values in some neighborhood of that point. A relative minimum is a point at which the function is less than or equal to its values in some neighborhood of the point. An inflection point is a point at which the derivative vanishes that is neither a relative manimum nor a relative minimum.

A single function can have any number of relative maxima, relative minima and inflection points.

Thanks, Dr. Rocket.
So, when my textbook says that for the graph of the equation y = 2x^3 + 3x^2 -36x+1 the maximum point is (-3, 82), this is only true if a neighbourhood relative to (-3,82) is specified?

Originally Posted by russell_c_cook

Thanks, Dr. Rocket.
So, when my textbook says that for the graph of the equation y = 2x^3 + 3x^2 -36x+1 the maximum point is (-3, 82), this is only true if a neighbourhood relative to (-3,82) is specified?

Yes. Think about the function. If x is very large in absolute value then the x^3 term dominates. It grows without bound as x becomes large and tends to minus infinity as x tends to minus infinity. So that function will have no global maximum or global minimum. It will have a local minimum at 2 and a local maximum at 3, which you find by locating the zeros of the derivative.

If your textbook actually said that the function has a maximum on the real line at
(-3,82) then your textbook is in error. That error strikes me as significant lack of understanding of basic theory on the part of the author. I would forgive that sort of thing coming from a freshman student, but not from the author of a textbook.

On the other hand if they were looking only on some sufficiently small interval containing the point (-3,82) then the statement may be correct in that context.

What textbook are you using ?

The textbook is from a photocopy my tutor gave me. I'll find out next time I see him and let you know. Thanks a lot for your clarification.