Question

Theory and ExamplesSketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

Accepted Answer

Start by understanding the characteristics of a differentiable function. A differentiable function is smooth and has no sharp corners or discontinuities. This means the function has a derivative at every point in its domain. Identify the points where the local maximum and minimum occur. The problem states that there is a local maximum at (1, 1) and a local minimum at (3, 3). At these points, the derivative of the function will be zero, indicating horizontal tangents. Consider the behavior of the function around the local maximum at (1, 1). As x approaches 1 from the left, the function should be increasing, and as x moves past 1 to the right, the function should be decreasing. This creates a peak at x = 1. Similarly, analyze the behavior around the local minimum at (3, 3). As x approaches 3 from the left, the function should be decreasing, and as x moves past 3 to the right, the function should be increasing. This creates a trough at x = 3. Sketch the graph by connecting these points smoothly, ensuring the function increases towards the local maximum, decreases after it, and then decreases towards the local minimum, increasing after it. The graph should reflect the smooth transition between these critical points, maintaining differentiability.

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

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Key Concepts

Differentiable Function

Local Maximum

Local Minimum