Exercise 2 Page 211

Let's start by reviewing what a turning point is.

A turning point is a point on the graph at which the function changes from increasing to decreasing or decreasing to increasing.

In the graph below, we can see the example function $f (x) = 2 x^{3} - 2 x,$ which has two turning points.

As we can see, a function with two turning points can intersect the $x - axis$ up to three times, with three different real zeros. In general a function with $n$ different real zeros has $n - 1$ turning points. Now that we know the relation between the number of zeros and turning points, let's recall the Fundamental Theorem of Algebra.

If f(x) is a polynomial of degree $n$ where $n > 0,$ then the equation $f (x) = 0$ has exactly $n$ solutions provided that each repeated solution is counted as many times as it repeats.

According to the theorem, a polynomial function of degree $n,$ where $n > 0,$ can have at most $n$ different real solutions. Therefore, it can have at most $n - 1$ turning points.