109. Suppose the derivative of the function y = f(x) is
y'=(x-1)^2(x-2).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection? (Hint: Draw the sign pattern for y'.)

Verified step by step guidance

1. Step 1: Identify critical points by setting the derivative y' = (x-1)^2(x-2) equal to zero. Solve for x to find the values where the slope of the tangent line is zero or undefined. This gives x = 1 and x = 2 as potential critical points.
2. Step 2: Analyze the sign pattern of y' by testing intervals around the critical points. Divide the x-axis into intervals based on the critical points: (-∞, 1), (1, 2), and (2, ∞). Choose test points within each interval and substitute them into y' to determine whether y' is positive or negative in each interval.
3. Step 3: Determine the behavior of the graph at x = 1 and x = 2. For x = 1, note that (x-1)^2 is always non-negative, so y' does not change sign at x = 1. This indicates that x = 1 is not a local maximum or minimum but could be a point of inflection. For x = 2, check if y' changes sign from positive to negative or vice versa to identify a local maximum or minimum.
4. Step 4: Investigate the second derivative y'' to confirm points of inflection. Compute y'' by differentiating y' = (x-1)^2(x-2) and analyze its sign changes. Points where y'' changes sign indicate points of inflection.
5. Step 5: Summarize the findings. Based on the sign pattern of y' and the behavior of y'' at x = 1 and x = 2, classify each critical point as a local minimum, local maximum, or point of inflection. Ensure the reasoning aligns with the sign changes observed in y' and y''.

Key Concepts

Derivative and Critical Points

The derivative of a function, denoted as y' or f'(x), represents the rate of change of the function at any point. Critical points occur where the derivative is zero or undefined, indicating potential locations for local maxima, minima, or points of inflection. Analyzing these points helps determine the behavior of the function around them.

Sign Chart Analysis

A sign chart is a visual tool used to determine the intervals where a function's derivative is positive, negative, or zero. By evaluating the sign of the derivative around critical points, one can infer whether the function is increasing or decreasing, which aids in identifying local maxima and minima, as well as points of inflection.

Local Extrema and Points of Inflection

Local extrema refer to points where a function reaches a local maximum or minimum, determined by the sign of the derivative. A point of inflection occurs where the concavity of the function changes, typically identified by a change in the sign of the second derivative. Understanding these concepts is crucial for analyzing the overall shape and behavior of the graph of the function.