Question

Describe the appearance of a smooth surface with a local maximum at a point.

Short answer

Question: Describe the appearance of a smooth surface with a local maximum at a point. Answer: The appearance of a smooth surface with a local maximum at a point will look like a small hill or a bump on the surface. The height of this hill is greater than its surrounding area but not necessarily the highest point on the entire surface. The transitions from lower to higher points will be gradual with no sharp edges or abrupt changes, making the surface appear smooth.

Step-by-step solution

1. Understanding Smooth Surfaces

A smooth surface is a surface that is continuously differentiable, which means it doesn't have any abrupt changes, sharp edges or corners. It can be visualized as a hill or a valley with smooth curves.

2. Understanding Local Maximum

A local maximum is a point on a surface where the value (height) of the function is greater than all the values of the function in its immediate neighborhood, but not necessarily the highest point on the entire surface. Imagine standing on a hill, where you're on a higher point than the area immediately surrounding you, but there might be higher hills elsewhere on the surface.

3. Appearance of a Smooth Surface with a Local Maximum

The appearance of a smooth surface with a local maximum at a point will be like a small bump or a hill on the surface. The height of the bump (or hill) will be greater than its neighboring area, but not necessarily the highest point on the entire surface. Since the surface is smooth, the transitions from low to high points will be gradual and have no sharp edges or corners. Picture tracing your finger along the surface, and the gradual increase in height as you approach the local maximum, followed by the gradual decrease as you move away from it.