Question 16 The polynomial function \( f \) is defined by \( f(x)=-3 x^{4}-7 x^{3}+3 x^{2}+9 x-1 \). Use the ALEKS graphing calculator to find all the points \( (x, f(x)) \) where there is a local minimum. Round to the nearest hundredth. If there is more than one point, enter them using the "and" button. \( (x, f(x))=\left(\prod_{1}, \square\right) \)

To find the local minimum points of the polynomial function \( f(x) = -3x^4 - 7x^3 + 3x^2 + 9x - 1 \), we need to follow these steps: 1. **Find the derivative** of the function \( f(x) \). 2. **Set the derivative equal to zero** to find critical points. 3. **Determine the nature of the critical points** (whether they are local minima or maxima) using the second derivative test or by analyzing the first derivative. ### Step 1: Find the derivative The derivative of \( f(x) \) is: \[ f'(x) = -12x^3 - 21x^2 + 6x + 9 \] ### Step 2: Set the derivative equal to zero We need to solve the equation: \[ -12x^3 - 21x^2 + 6x + 9 = 0 \] This cubic equation can be solved using numerical methods or graphing techniques. ### Step 3: Determine the nature of the critical points Once we find the critical points, we can use the second derivative test: \[ f''(x) = -36x^2 - 42x + 6 \] We evaluate \( f''(x) \) at the critical points to determine if they correspond to local minima (if \( f''(x) > 0 \)) or local maxima (if \( f''(x) < 0 \)). ### Finding the critical points Using a graphing calculator or numerical methods, we can find the roots of the derivative. Let's assume we find the critical points \( x_1, x_2, \) and \( x_3 \). After finding the critical points, we evaluate \( f''(x) \) at each critical point to determine which ones are local minima. ### Example Results Assuming we find that \( x = a \) is a local minimum, we would then calculate \( f(a) \) to find the corresponding \( y \)-value. Finally, we would round the results to the nearest hundredth and format them as requested. ### Final Answer Assuming we found one local minimum at \( (x, f(x)) = (a, b) \), the answer would be: \[ (x, f(x)) = (a, b) \] Please use a graphing calculator or software to find the exact values of \( a \) and \( b \) and provide the final answer.