\begin{tabular}{l} (iv) From the graph, estimate the co-ordinates of the local maximum point and the local \\ minimum point of \( g(x) \), as accurately as possible. \\ Local maximum point: \\ Local minimum point: \\ different function, \( h(x) \), has the following equation: \\ Find \( h^{\prime}(x) \), the derivative of \( h(x) \) \\ \( h^{\prime}(x)=x^{3}+2 x^{2}-x-8 \) \\ \hline (x) \end{tabular}

It looks like you're working on a multi-part problem involving functions \( g(x) \) and \( h(x) \). I'll address each part based on the information provided: --- ### (iv) Estimating Local Maximum and Minimum Points of \( g(x) \) **Task:** From the graph, estimate the coordinates of the local maximum and minimum points of \( g(x) \). **Answer:** To accurately estimate the coordinates of the local maximum and minimum points of \( g(x) \), you would need to analyze the graph of the function. Here's how you can approach it: 1. **Identify Peaks and Troughs:** - **Local Maximum:** Look for the highest point (peak) in a specific interval. - **Local Minimum:** Look for the lowest point (trough) in a specific interval. 2. **Estimate Coordinates:** - **Local Maximum Point:** Determine the \( x \)-value where the peak occurs and read the corresponding \( y \)-value. - **Local Minimum Point:** Determine the \( x \)-value where the trough occurs and read the corresponding \( y \)-value. **Example Format:** - **Local maximum point:** \( (x_{\text{max}}, y_{\text{max}}) \) - **Local minimum point:** \( (x_{\text{min}}, y_{\text{min}}) \) *Since the graph isn't provided here, you'll need to apply this method using your specific graph to find the estimated coordinates.* --- ### Finding the Derivative \( h^{\prime}(x) \) of \( h(x) \) **Given:** You mentioned that for a different function \( h(x) \), the derivative is provided as: \[ h^{\prime}(x) = x^{3} + 2x^{2} - x - 8 \] **Task:** Find \( h^{\prime}(x) \), the derivative of \( h(x) \). **Answer:** If \( h^{\prime}(x) = x^{3} + 2x^{2} - x - 8 \) is provided, it implies that the derivative has already been calculated. However, if you're looking to verify or understand how to find the derivative, here's a step-by-step guide assuming you have the original function \( h(x) \): 1. **Original Function \( h(x) \):** Suppose \( h(x) \) is a polynomial function. For example: \[ h(x) = \frac{1}{4}x^{4} + \frac{2}{3}x^{3} - \frac{1}{2}x^{2} - 8x + C \] *(Note: The constant \( C \) represents the integration constant and would disappear upon differentiation.)* 2. **Differentiation Rules Applied:** - **Power Rule:** \(\frac{d}{dx} [x^n] = n \cdot x^{n-1}\) - **Constant Multiple Rule:** \(\frac{d}{dx} [k \cdot f(x)] = k \cdot f^{\prime}(x)\) - **Sum Rule:** \(\frac{d}{dx} [f(x) + g(x)] = f^{\prime}(x) + g^{\prime}(x)\) 3. **Applying to \( h(x) \):** \[ h^{\prime}(x) = \frac{d}{dx} \left( \frac{1}{4}x^{4} \right) + \frac{d}{dx} \left( \frac{2}{3}x^{3} \right) - \frac{d}{dx} \left( \frac{1}{2}x^{2} \right) - \frac{d}{dx} (8x) + \frac{d}{dx} (C) \] \[ h^{\prime}(x) = x^{3} + 2x^{2} - x - 8 \] This confirms the provided derivative. **Conclusion:** If you have the original function \( h(x) \), you can apply these differentiation rules to find \( h^{\prime}(x) \). Since \( h^{\prime}(x) \) is already given as \( x^{3} + 2x^{2} - x - 8 \), it appears the derivative has been correctly determined. --- If you need further assistance with specific parts of the problem or have additional information (like the equation for \( h(x) \) or the graph of \( g(x) \)), feel free to provide more details!