Find the critical points of f(x) = x³ - 3x² + 2 and sketch the graph showing all local maxima, minima, and inflection points.

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To find the critical points of f(x) equals x cubed minus 3x squared plus 2, we first need to find the derivative. The derivative of f(x) is 3x squared minus 6x. To find the critical points, we set the derivative equal to zero and solve for x. Factoring out 3x, we get 3x times (x minus 2) equals zero. This gives us x equals 0 or x equals 2. These are our critical points. When we evaluate the function at these points, we get the critical points (0, 2) and (2, -2). Looking at the graph, we can see that (0, 2) is a local maximum because the derivative changes from positive to negative, and (2, -2) is a local minimum because the derivative changes from negative to positive. Now, let's classify our critical points. For x equals 0, the derivative changes from positive to negative, which means we have a local maximum at the point (0, 2). For x equals 2, the derivative changes from negative to positive, indicating a local minimum at the point (2, -2). To find inflection points, we need to calculate the second derivative. The second derivative of our function is 6x minus 6. Setting this equal to zero gives us x equals 1. When we evaluate the function at x equals 1, we get f(1) equals 0, so (1, 0) is our inflection point. At this point, the concavity of the function changes from concave down to concave up. The tangent line at the inflection point passes through the curve, with the curve changing from lying above the tangent to lying below it. Now, let's analyze the intervals of increase and decrease for our function. The derivative f prime of x equals 3x squared minus 6x, which can be factored as 3x times (x minus 2). The function is increasing when the derivative is positive, which occurs when x is less than 0 or x is greater than 2. The function is decreasing when the derivative is negative, which happens when x is between 0 and 2. Next, let's analyze the concavity. The second derivative f double prime of x equals 6x minus 6, which equals 6 times (x minus 1). The function is concave down when the second derivative is negative, which occurs when x is less than 1. The function is concave up when the second derivative is positive, which happens when x is greater than 1. Finally, let's analyze the end behavior. As x approaches negative infinity, f(x) approaches negative infinity. As x approaches positive infinity, f(x) approaches positive infinity. This completes our analysis of the function f(x) equals x cubed minus 3x squared plus 2. Let's summarize our complete analysis of the function f(x) equals x cubed minus 3x squared plus 2. We found two critical points: a local maximum at (0, 2) and a local minimum at (2, -2). We identified an inflection point at (1, 0), where the concavity changes. The function is increasing when x is less than 0 or x is greater than 2, and it's decreasing when x is between 0 and 2. The function is concave down when x is less than 1 and concave up when x is greater than 1. Looking at our graph, we can see all these features clearly. The curve rises to a maximum at (0, 2), then falls to a minimum at (2, -2), and then rises again. The inflection point at (1, 0) marks where the curve changes from bending downward to bending upward. This complete analysis gives us a thorough understanding of the behavior of our cubic function. To summarize what we've learned: The cubic function f(x) equals x cubed minus 3x squared plus 2 has two critical points where the derivative equals zero. At x equals 0, we have a local maximum with function value 2. At x equals 2, we have a local minimum with function value negative 2. The function has an inflection point at x equals 1, where the second derivative equals zero and the concavity changes. The function increases when x is less than 0 or greater than 2, and decreases when x is between 0 and 2. This analysis demonstrates how calculus techniques, particularly derivatives, allow us to completely understand and sketch the behavior of a function.