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Answer :
To find the number of turning points for the given function [tex]\( C(x) = x^9 + 2x^4 + 4x^8 - 5x \)[/tex], we need to consider the degree of the polynomial.
### Steps to Find the Maximum Number of Turning Points:
1. Identify the Degree of the Polynomial:
- Look at the term with the highest exponent, which is [tex]\( x^9 \)[/tex]. So, the degree of the polynomial is 9.
2. Determine the Maximum Number of Turning Points:
- For any polynomial function, the maximum number of turning points is one less than the degree of the polynomial.
- Since the degree of [tex]\( C(x) \)[/tex] is 9, the maximum number of turning points is [tex]\( 9 - 1 = 8 \)[/tex].
Therefore, the function [tex]\( C(x) = x^9 + 2x^4 + 4x^8 - 5x \)[/tex] has at most 8 turning points.
### Steps to Find the Maximum Number of Turning Points:
1. Identify the Degree of the Polynomial:
- Look at the term with the highest exponent, which is [tex]\( x^9 \)[/tex]. So, the degree of the polynomial is 9.
2. Determine the Maximum Number of Turning Points:
- For any polynomial function, the maximum number of turning points is one less than the degree of the polynomial.
- Since the degree of [tex]\( C(x) \)[/tex] is 9, the maximum number of turning points is [tex]\( 9 - 1 = 8 \)[/tex].
Therefore, the function [tex]\( C(x) = x^9 + 2x^4 + 4x^8 - 5x \)[/tex] has at most 8 turning points.
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