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

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** ---