Determine the end behavior and number of turning points of each polynomial function. $$ \begin{array}{l} ext { 3. } f(x)=x^{3}-14 x-4 \ ext { 4. } g(x)=5-17 x^{7}+9 x^{10}\end{array} $$

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Determine the end behavior and number of turning points of each polynomial function. $$ \begin{array}{l}	ext { 3. } f(x)=x^{3}-14 x-4 \ 	ext { 4. } g(x)=5-17 x^{7}+9 x^{10}\end{array} $$

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Let's analyze each polynomial function to determine their end behavior and the number of turning points. --- ### **3. $$ f(x) = x^{3} - 14x - 4 $$** **a. End Behavior:** - **Degree:** The polynomial is of degree **3**, which is **odd**. - **Leading Coefficient:** The coefficient of $$ x^3 $$ is **1** (positive). For an odd-degree polynomial with a positive leading coefficient: - **As $$ x \to +\infty $$, $$ f(x) \to +\infty $$** - **As $$ x \to -\infty $$, $$ f(x) \to -\infty $$** **b. Number of Turning Points:** - **Maximum Turning Points:** For a polynomial of degree $$ n $$, the maximum number of turning points is $$ n-1 $$. Here, $$ 3-1 = 2 $$. - **Verification:** - **First Derivative:** $$ f'(x) = 3x^2 - 14 $$ - **Critical Points:** Set $$ f'(x) = 0 $$: $$ 3x^2 - 14 = 0 \implies x^2 = \frac{14}{3} \implies x = \pm \sqrt{\frac{14}{3}} $$ - There are **two distinct real critical points**, indicating **two turning points**. **Summary for $$ f(x) $$:** - **End Behavior:** - $$ x \to +\infty \Rightarrow f(x) \to +\infty $$ - $$ x \to -\infty \Rightarrow f(x) \to -\infty $$ - **Turning Points:** **2** --- ### **4. $$ g(x) = 5 - 17x^{7} + 9x^{10} $$** **a. End Behavior:** - **Degree:** The polynomial is of degree **10**, which is **even**. - **Leading Coefficient:** The coefficient of $$ x^{10} $$ is **9** (positive). For an even-degree polynomial with a positive leading coefficient: - **As $$ x \to +\infty $$, $$ g(x) \to +\infty $$** - **As $$ x \to -\infty $$, $$ g(x) \to +\infty $$** **b. Number of Turning Points:** - **Maximum Turning Points:** For a polynomial of degree $$ n $$, the maximum number of turning points is $$ n-1 = 10-1 = 9 $$. - **Determining the Actual Number:** - **First Derivative:** $$ g'(x) = 90x^{9} - 119x^{6} $$ - **Critical Points:** Set $$ g'(x) = 0 $$: $$ 90x^{9} - 119x^{6} = x^{6}(90x^{3} - 119) = 0 $$ - $$ x = 0 $$ (with multiplicity **6**) and $$ x = \left(\frac{119}{90}\right)^{1/3} $$ - **Analyzing Sign Changes:** - For $$ x < \left(\frac{119}{90}\right)^{1/3} $$, $$ g'(x) < 0 $$ (function is decreasing). - At $$ x = \left(\frac{119}{90}\right)^{1/3} $$, $$ g'(x) = 0 $$ (local minimum). - For $$ x > \left(\frac{119}{90}\right)^{1/3} $$, $$ g'(x) > 0 $$ (function is increasing). - Therefore, there is **only one real turning point**. **Summary for $$ g(x) $$:** - **End Behavior:** - $$ x \to +\infty \Rightarrow g(x) \to +\infty $$ - $$ x \to -\infty \Rightarrow g(x) \to +\infty $$ - **Turning Points:** **1** --- ### **Final Summary:** 1. **For $$ f(x) = x^{3} - 14x - 4 $$:** - **End Behavior:** - $$ x \to +\infty \Rightarrow f(x) \to +\infty $$ - $$ x \to -\infty \Rightarrow f(x) \to -\infty $$ - **Number of Turning Points:** **2** 2. **For $$ g(x) = 5 - 17x^{7} + 9x^{10} $$:** - **End Behavior:** - $$ x \to +\infty \Rightarrow g(x) \to +\infty $$ - $$ x \to -\infty \Rightarrow g(x) \to +\infty $$ - **Number of Turning Points:** **1** --- For $$ f(x) = x^{3} - 14x - 4 $$: - **End Behavior:** As $$ x \to +\infty $$, $$ f(x) \to +\infty $$; as $$ x \to -\infty $$, $$ f(x) \to -\infty $$. - **Number of Turning Points:** 2. For $$ g(x) = 5 - 17x^{7} + 9x^{10} $$: - **End Behavior:** As $$ x \to +\infty $$ and $$ x \to -\infty $$, $$ g(x) \to +\infty $$. - **Number of Turning Points:** 1.

Determine the end behavior and number of turning points of each polynomial function. \( \begin{array}{l}\text { 3. } f(x)=x^{3}-14 x-4 \\ \text { 4. } g(x)=5-17 x^{7}+9 x^{10}\end{array} \)

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