Questions: Determine the maximum possible number of turning points for the graph of the function. f(x)=x^5(x^5+3)(3x+5) A. 11 B. 5 C. 10 D. 30

Determine the maximum possible number of turning points for the graph of the function. f(x)=x^5(x^5+3)(3x+5) A. 11 B. 5 C. 10 D. 30
Transcript text: Determine the maximum possible number of turning points for the graph of the function. \[ f(x)=x^{5}\left(x^{5}+3\right)(3 x+5) \] A. 11 B. 5 C. 10 D. 30
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Solution

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Solution Steps

To determine the maximum possible number of turning points for the graph of a polynomial function, we need to consider the degree of the polynomial. The maximum number of turning points is one less than the degree of the polynomial. First, expand the given function to find its degree.

Step 1: Determine the Function

The given function is

\[ f(x) = x^{5}(x^{5} + 3)(3x + 5). \]

Step 2: Expand the Function

Upon expanding the function, we find:

\[ f(x) = 3x^{11} + 5x^{10} + 9x^{6} + 15x^{5}. \]

Step 3: Find the Degree of the Polynomial

The degree of the polynomial \( f(x) \) is \( 11 \).

Step 4: Calculate Maximum Turning Points

The maximum number of turning points is given by

\[ \text{Maximum Turning Points} = \text{Degree} - 1 = 11 - 1 = 10. \]

Final Answer

The maximum possible number of turning points for the graph of the function is \\(\boxed{10}\\).

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