Imagine yourself traveling along the graph of a polynomial, moving from left
to right. Sometimes you go ‘uphill’, sometimes
‘downhill’, and sometimes you change direction. Such a change of direction is called a turning point. The purpose of this section is to make this concept more precise. 

Roughly, a local maximum is a point on a curve where there
is a ‘local high spot’. That is, if you ‘stand on’ the point and don't look too far away, then everything you see is lower (or at the same height). If you look too far away (as the red arrow indicates, at right) then you may see points that are higher. People often say ‘local max’ instead of ‘local maximum’, for brevity. (The precise definition of a local max is a bit complicated, and is usually covered in a Calculus course.) There is an analogous description for a local minimum (‘local min’, for short). 

The following fact is easily proved in a Calculus course:
A polynomial of degree $n$ can have at most $n1$ turning points.
The polynomial $\,y = x^3\,$ has no turning points. It has a horizontal tangent line at $\,(0,0)\,$ which is not a turning point. 

no turning points 
The polynomial $\,y = x^3  x\,$ has two turning points. 

two turning points 
Does there exist a cubic polynomial with exactly one turning point? Analysis of end behavior shows that the answer is NO, as follows: If the leading coefficient of a cubic polynomial is positive, then: as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty$ as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty$ If the leading coefficient of a cubic polynomial is negative, then: as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty$ as $\,x\rightarrow\infty\,$, $\,y\rightarrow \infty$ In both cases, one ‘end’ is going to infinity, and the other to negative infinity. If a polynomial turns exactly once, then both the righthand and lefthand end behaviors must be the same. Hence, a cubic polynomial cannot have exactly one turning point. 
with exactly one turning point 
Calculus results about derivatives, together with the Fundamental Theorem of Algebra,
will eventually firm up the concepts in this section.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
