The function \( f(x) = x^3 - 3x^2 + 4 \) has a local minimum at \( x = a \). Find the value of \( a \).
["Title: Finding the Local Minimum of ( f(x) = x^3 - 3x^2 + 4 ): Step-by-Step Calculation", "Understanding the critical points and local extrema of a function is a fundamental skill in calculus. In this article, we explore how to determine where the function ( f(x) = x^3 - 3x^2 + 4 ) achieves a local minimum—and specifically, find the value ( a ) at which this minimum occurs.", "---", "### What is a Local Minimum?", "A local minimum of a function ( f(x) ) at ( x = a ) is a point where ( f(a) ) is less than or equal to values of ( f(x) ) near ( x = a ). To locate local minima, we typically:", "1. Find the first derivative ( f'(x) ) and solve ( f'(x) = 0 ) to identify critical points.\n2. Use the second derivative test or analyze the sign change of ( f'(x) ) around critical points to classify them.\n3. Confirm the nature of each critical point as a local minimum, maximum, or neither.", "---", "### Step 1: Compute the First Derivative", "Given:\n[\nf(x) = x^3 - 3x^2 + 4\n]", "Differentiate to find ( f'(x) ):\n[\nf'(x) = \frac{d}{dx}(x^3 - 3x^2 + 4) = 3x^2 - 6x\n]", "---", "### Step 2: Find Critical Points", "Set ( f'(x) = 0 ):\n[\n3x^2 - 6x = 0\n]\nFactor out ( 3x ):\n[\n3x(x - 2) = 0\n]", "Solve for ( x ):\n[\nx = 0 \quad \ ext{or} \quad x = 2\n]", "These are the critical points. Next, we determine which (if any) is a local minimum.", "---", "### Step 3: Use the Second Derivative Test", "Compute the second derivative ( f''(x) ):\n[\nf''(x) = \frac{d}{dx}(3x^2 - 6x) = 6x - 6\n]", "Evaluate ( f''(x) ) at each critical point:", "- At ( x = 0 ):\n [\n f''(0) = 6(0) - 6 = -6 < 0 \quad \Rightarrow \quad x = 0 \ ext{ is a local maximum}\n ]", "- At ( x = 2 ):\n [\n f''(2) = 6(2) - 6 = 12 - 6 = 6 > 0 \quad \Rightarrow \quad x = 2 \ ext{ is a local minimum}\n ]", "---", "### Conclusion: The Local Minimum Occurs at ( x = 2 )", "Thus, the function ( f(x) = x^3 - 3x^2 + 4 ) has a local minimum at ( x = a = 2 ). This result helps in optimization problems and understanding function behavior—key aspects for students, educators, and professionals applying calculus.", "---", "### Why This Matters", "Identifying local minima enables better modeling of real-world phenomena such as cost minimization, profit maximization, and physical systems. Mastery of derivatives and critical point analysis is essential in engineering, economics, data science, and beyond.", "Keywords: ( f(x) = x^3 - 3x^2 + 4 ), local minimum, critical points, derivative test, calculus, optimization, ( f'(x) = 3x^2 - 6x ), ( f''(x) = 6x - 6 )", "---", "If you're studying calculus or preparing for exams, recognizing how to analyze derivatives and classify critical points is crucial. Always verify your results using second derivative tests or sign changes in first derivatives—practical strategies for accuracy."]
