Answer :
Consider the polynomial function
[tex]$$
f(x) = x^9 - 4x^7 + 6x - 3.
$$[/tex]
Since this is a polynomial of degree 9, we can determine the following:
1. Maximum number of real zeros:
A polynomial of degree [tex]$n$[/tex] can have at most [tex]$n$[/tex] real zeros. Here, the degree is 9, so the maximum number of real zeros is
[tex]$$
9.
$$[/tex]
2. Maximum number of [tex]$x$[/tex]-intercepts:
The [tex]$x$[/tex]-intercepts of the graph are the points where [tex]$f(x)=0$[/tex]. These are exactly the real zeros of the function. Therefore, the maximum number of [tex]$x$[/tex]-intercepts is also
[tex]$$
9.
$$[/tex]
3. Maximum number of turning points:
A polynomial of degree [tex]$n$[/tex] can have at most [tex]$n-1$[/tex] turning points (local maxima and minima). For a polynomial of degree 9, the maximum number of turning points is
[tex]$$
9 - 1 = 8.
$$[/tex]
Thus, the answers are:
- The function has a maximum of [tex]$\boxed{9}$[/tex] real zeros.
- The function has a maximum of [tex]$\boxed{9}$[/tex] [tex]$x$[/tex]-intercepts.
- The function has a maximum of [tex]$\boxed{8}$[/tex] turning points.
[tex]$$
f(x) = x^9 - 4x^7 + 6x - 3.
$$[/tex]
Since this is a polynomial of degree 9, we can determine the following:
1. Maximum number of real zeros:
A polynomial of degree [tex]$n$[/tex] can have at most [tex]$n$[/tex] real zeros. Here, the degree is 9, so the maximum number of real zeros is
[tex]$$
9.
$$[/tex]
2. Maximum number of [tex]$x$[/tex]-intercepts:
The [tex]$x$[/tex]-intercepts of the graph are the points where [tex]$f(x)=0$[/tex]. These are exactly the real zeros of the function. Therefore, the maximum number of [tex]$x$[/tex]-intercepts is also
[tex]$$
9.
$$[/tex]
3. Maximum number of turning points:
A polynomial of degree [tex]$n$[/tex] can have at most [tex]$n-1$[/tex] turning points (local maxima and minima). For a polynomial of degree 9, the maximum number of turning points is
[tex]$$
9 - 1 = 8.
$$[/tex]
Thus, the answers are:
- The function has a maximum of [tex]$\boxed{9}$[/tex] real zeros.
- The function has a maximum of [tex]$\boxed{9}$[/tex] [tex]$x$[/tex]-intercepts.
- The function has a maximum of [tex]$\boxed{8}$[/tex] turning points.
