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