Theory and Examples


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

Verified step by step guidance

1. Start by understanding the concept of local minima. A local minimum is a point where the function value is lower than all nearby points. This means the derivative at these points is zero, indicating a horizontal tangent line.
2. Identify the given local minima points: (1, 1) and (3, 3). These points suggest that the function has a 'valley' or 'dip' at these coordinates.
3. Consider the behavior of the function around these points. Since the function is differentiable, it is smooth and continuous. The derivative changes sign around these minima points, from negative to positive.
4. Sketch the graph by plotting the points (1, 1) and (3, 3) on a coordinate plane. Draw a curve that dips down at these points, ensuring the tangent is horizontal at these minima.
5. Ensure the graph is smooth and continuous, reflecting the differentiability of the function. Between the minima, the graph should rise, and outside these points, it should fall, creating a 'U' shape around each minimum.

Key Concepts

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. This means the function is smooth and continuous, without any sharp corners or cusps. Understanding differentiability is crucial for sketching graphs, as it ensures the function's behavior can be predicted using its derivative.

Local Minima

A local minima of a function is a point where the function value is lower than all other nearby points. At a local minima, the derivative of the function changes from negative to positive, indicating a 'valley' in the graph. Identifying local minima is essential for sketching the graph accurately, as it highlights where the function reaches its lowest points locally.

Graph Sketching

Graph sketching involves plotting the general shape of a function based on its critical points, such as local minima and maxima, and its behavior at infinity. It requires understanding the function's derivative to determine where the function is increasing or decreasing, and how it curves. This skill is vital for visualizing the function's behavior and ensuring the graph reflects all given conditions.