To determine if the value 3 is an upper bound for the zeros of the polynomial function [tex]\( f(x) = -3x^3 + 20x^2 - 36x + 16 \)[/tex], we use Descartes' Rule of Signs. This rule helps us examine the possible number of positive and negative real zeros of a polynomial based on the number of sign changes in the coefficients of the polynomial.
### Step-by-Step Solution:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\( f(x) = -3x^3 + 20x^2 - 36x + 16 \)[/tex] has the following coefficients:
[tex]\[
-3, 20, -36, 16
\][/tex]
2. Analyze the sign changes:
We inspect the coefficients sequentially to count how often the sign changes from one coefficient to the next:
- From [tex]\(-3\)[/tex] to [tex]\(20\)[/tex] (negative to positive), there is a sign change.
- From [tex]\(20\)[/tex] to [tex]\(-36\)[/tex] (positive to negative), there is a sign change.
- From [tex]\(-36\)[/tex] to [tex]\(16\)[/tex] (negative to positive), there is a sign change.
3. Count the sign changes:
In total, there are three sign changes.
4. Apply Descartes' Rule of Signs:
According to Descartes' Rule of Signs, the number of positive real zeros of the polynomial is equal to the number of sign changes in the coefficients, or less than that by an even number. Thus, the possible number of positive real zeros for this polynomial could be [tex]\(3, 1,\)[/tex] or [tex]\(0\)[/tex].
5. Interpret the result:
Since there are possible positive real zeros (the count is 3 or 1), the value 3 cannot be guaranteed to be an upper bound for the zeros of the function. An upper bound would imply that there are no positive real zeros greater than 3, but we have evidence that there could be positive real zeros.
### Conclusion:
Given the analysis of the sign changes and the application of Descartes' Rule of Signs, the value 3 is not conclusively an upper bound for the zeros of the polynomial [tex]\( f(x) = -3x^3 + 20x^2 - 36x + 16 \)[/tex].
Therefore, the statement "The value 3 is an upper bound for the zeros of the function [tex]\( f(x) = -3x^3 + 20x^2 - 36x + 16 \)[/tex]" is:
B. False
