Question

Choose a point P on a smooth curve C in the plane.

1) P is a maximum point on the curve.
2) P is a minimum point on the curve.
3) P is an inflection point on the curve.
4) P is a point of tangency on the curve.

Answer 1

A point on a curve can be classified as a maximum point, a minimum point, an inflection point, or a point of tangency.

Step-by-step explanation:

The point P on a smooth curve C in the plane can be classified into four possibilities:

P is a maximum point on the curve if the curve has a local maximum at point P. This means that the curve is increasing on one side of P and decreasing on the other side.

P is a minimum point on the curve if the curve has a local minimum at point P. This means that the curve is decreasing on one side of P and increasing on the other side.

P is an inflection point on the curve if the curvature of the curve changes sign at point P. This means that the curve changes from being concave up to concave down, or vice versa.

P is a point of tangency on the curve if the curve is tangent to a line at point P. This means that the slope of the curve at P is equal to the slope of the line.