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?

Re: Maximum and minimum points of a curve

Quote:

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.