Difference between Convex and Concave Curves

Key Difference: A concave curve is rounded inward, whereas a convex curve is rounded like the exterior of a sphere.

A curve is very different from a straight line. A curve has a varying slope. It is a wiggly line or bent line which wiggles or bends to join any two points on a graph or a map. Curves can be divided into categories of convex and concave curves. A concave curve rounds inward. On the other hand, a convex curve is rounded like the exterior of a sphere or a circle. Many people understand the terms by considering that a concave curve is similar to a valley and a convex curve is similar to a mountain. Thus, it helps in remembering the differences between the two.

In a concave curve, a straight line connecting any two points on the curve lies entirely under the curve. However, in a convex curve, a straight line joining any two points lies totally above the curve. Both the curves are regarded opposite to each other. Therefore, a negative convexity refers to a term named as concavity. Concavity and inflection point describe the directions of a curve. Concavity describes the way that a curve bends. An inflection point is a point where the function has a tangent and the concavity changes. For example, if a curve is concave down (simply concave) then the graph of the curve is bent down, otherwise for the case of a concave up (convex) type of curve, the graph of the curve is bent upward. Convex and concave are often used to denote a gentle and subtle curve that is generally found in mirrors and lens.

Comparison between Convex and Concave Curves:

 

Convex Curves

Concave Curves

Definition

A convex curve is rounded like the exterior of a sphere.

A concave curve is rounded inward.

Analogy

Mountain

Valley

Directionality of a curve

Concave upward - if the curve ‘bends’ upward

Concave downward - if the curve ‘bends’ downward

Theorem

If a function is concave up on a given open interval, then it is also continuous on it.

(Let f be a real function which is differentiable on the open interval (a..b).)

f is convex on (a..b) iff its second derivative D2f is ≥0 on (a..b)

If a function is concave down on a given open interval, then it is also continuous on it.

(Let f be a real function which is differentiable on the open interval (a..b).)

f is concave on (a..b) iff its second derivative D2f is ≤0 on (a..b).

Example

Indifference curves are convex curves with respect to the origin.

The inside of the spoon.

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Comments

I think you need to swap the mountain and valley. Isn't the valley equivalent concave up or convex and the mountain concave down or strictly concave?

WOW THIS IS SO AMAZING SO HELPFUL AND SO INFORMATIVE THIS HELPED MY KIDS DO THEIR ASSIGNMENTS IN THEIR MAPE AND MATH SUBJECT OMG

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