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Matched Problem 5 Follow the graphing strategy and analyze the function \( f(x)=x^{3}-3 x^{2} \)

Ask by Ruiz Burns. in the United States
Mar 10,2025

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To analyze the function \( f(x) = x^3 - 3x^2 \), follow these steps: 1. **Find Intercepts:** - **\(x\)-intercepts:** \( x = 0 \) (double root) and \( x = 3 \). - **\(y\)-intercept:** \( (0,0) \). 2. **Determine End Behavior:** - As \( x \to \infty \), \( f(x) \to \infty \). - As \( x \to -\infty \), \( f(x) \to -\infty \). 3. **Find Critical Points:** - **Local Maximum:** \( (0,0) \). - **Local Minimum:** \( (2,-4) \). 4. **Find Inflection Point:** - **Inflection Point:** \( (1,-2) \). 5. **Concavity:** - Concave downward for \( x < 1 \). - Concave upward for \( x > 1 \). **Summary:** - The graph has a local maximum at \( (0,0) \), a local minimum at \( (2,-4) \), and an inflection point at \( (1,-2) \). - It crosses the \( x \)-axis at \( x = 0 \) and \( x = 3 \), and the \( y \)-axis at \( (0,0) \). - The function increases from \( -\infty \) to \( \infty \), with changes in concavity at \( x = 1 \).

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Beyond the Answer

To analyze the function \( f(x) = x^{3} - 3x^{2} \), let's start by finding its critical points. First, we take the derivative \( f'(x) = 3x^{2} - 6x \) and set it to zero to find where the function's slope is zero. Factoring gives \( f'(x) = 3x(x - 2) \), leading to critical points at \( x = 0 \) and \( x = 2 \). Evaluating the function at these points reveals local behavior, helping you determine maxima, minima, and inflection points. Now let's explore how this function might apply to the real-world! Imagine you're analyzing a production function for a company: \( f(x) \) could represent profit based on the number of units produced. By studying the critical points, you can find optimal production levels – minimizing costs and maximizing profits. Understanding \( f(x) \) in this way allows businesses to strategically navigate their operations!

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