The curve with an upward opening is called concave or convex, and its shape is ∨.
A curve with a downward opening is called a concave curve or a convex curve, and its shape is ∩.
Mathematically, upper concavity, lower concavity, upper convexity and lower convexity are collectively called convexity of curve, which is a graphic style in plane coordinate system. In fact, it can be divided into convex and concave. From the tangent point of view, the tangent of any point on the lower convex arc is below the curved arc, and the tangent of any point on the upper convex arc is above the curved arc.
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meaning
When studying the change of function graph, only studying monotonicity can not fully reflect its changing law.
Although the function monotonically increases in the interval [a, b], there are different bending conditions. From left to right, the curve is concave downward at first, then changes the bending direction after passing point P, and the curve is convex upward.
Therefore, when studying the graph of a function, we should not only study its monotonicity, but also study its bending direction and its changing points.
When the curve is concave downward, the arc segment of the curve is located above the tangent of any point on the arc segment, and when the curve is convex upward, the arc segment of the curve is located below the tangent of any point on the arc segment.
How to judge concavity and convexity with curvature circle
The concavity and convexity of the function is expressed by twice derivation of curvature circle and twice derivation of invariant sign, which shows that concavity and convexity are invariant. Curvature circle, also called osculating circle. On the normal line of a point M on the curve, take a point D on the concave side, make DM equal to the radius of curvature of this point, and make a circle with D as the center and DM as the radius. This circle is called the curvature circle of the curve at point m; Near point m, the curvature arc is very close to the curve arc, so the curvature circle is also called the osculating circle.
The extreme point is the point where the concave-convex changes, and the sign of the derivative remains unchanged twice, indicating that the concave-convex remains unchanged and the extreme point does not exist; Whether there is a zero point depends on the curvature circle to judge the concavity and convexity; X+y=2 means that the curvature of a circle with the origin as the center and the radius of root number 2 is the radius, and the curve of 10 is a concave curve; The concave curve must have an intersection with the coordinate axis, that is, the function has a zero point; The curvature circle has the following properties: the curvature circle and the curve have the same tangent and curvature at point m;
There is the same concave direction as the curve near point m; Therefore, in practical engineering design problems, a circular arc with a curvature circle near point M is often used to approximately replace the curved circular arc to simplify the problem.