# Difference between concave and convex

The difference between concave and convex can be explained as follows → The term convex refers to the fact that a surface has an inward curvature, while if it were concave the curvature would be outward. Thus, we can describe it in another way. The central part of a concave surface is more depressed or depressed. On the other hand, if it were convex, that central part would show a prominence.

To understand it better we can cite some examples. First, the classic case of a sphere, whose surface is convex. However, if we cut it in two and keep the lower half, we would have a concave object, with a sag (assuming the interior of the sphere is empty).

Another example of convex would be a mountain, since it is a prominence with respect to the earth’s surface. On the contrary, a well is concave, since entering it implies sinking, below the level of the earth’s surface.

It should also be noted that to define an object as concave or convex perspective must also be taken into account. Thus, a bowl of soup, for example, when it is ready to serve, is concave, it has a sag. However, if we turn it over, the plate will be convex.

On the other hand, in the case of parabolas, they are convex if they have a U shape, but concave if they have an inverted U shape.

## Concave and convex functions

If the second derivative of a function is less than zero at a point, then the function is concave at that point. On the other hand, if it is greater than zero, it is convex at that point. The above can be expressed as follows:

If f »(x) <0, f (x), it is concave.

If f »(x)> 0, f (x) it is convex.

For example, in the equation f (x) = x 2 + 5x-6, we can calculate its first derivative:

f ‘(x) = 2x + 5

Then we find the second derivative:

f »(x) = 2

Therefore, since f »(x) is greater than 0, the function is convex for every value of x, as we see in the graph below:

Now, let’s see the case of this other function: f (x) = – 4x 2 + 7x + 9.

f ‘(x) = – 8x + 7

f »(x) = – 8

Therefore, since the second derivative is less than 0, the function is concave for every value of x.

But now let’s look at the following equation: -5 x 3 + 7x 2 +5 x-4

f ‘(x) = – 15x 2 + 14x + 5

f »(x) = – 30x + 14

We set the second derivative equal to zero:

-30x + 14 = 0

x = 0.4667

So when x is greater than 0.4667, f »(x) is greater than zero, so the function is convex. While if x is less than 0.4667, the function is concave, as we see in the graph below:

## Convex and concave polygon

A convex polygon is one where two of its points can be joined, drawing a straight line that remains within the figure. Likewise, its interior angles are all less than 180º.

On the other hand, a concave polygon is one where, to join two of its points, a straight line must be drawn that is outside the figure, this being an exterior diagonal that joins two vertices. Furthermore, at least one of its interior angles is greater than 180º.

We can see a comparison in the image below: