\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}

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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:

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### (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.*

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### 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.

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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!
To estimate the local maximum and minimum points of $$ g(x) $$, analyze the graph to find the highest and lowest points, then note their coordinates. For $$ h(x) $$, the derivative $$ h^{\prime}(x) $$ is already given as $$ x^{3} + 2x^{2} - x - 8 $$.

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