The term concave is used to describe a surface that has an inward curvature, its central part being the most depressed or depressed.
Therefore, we say, for example, that an excavation towards the subsoil is concave.
Likewise, it is possible to analyze whether there are geometric figures that are also concave. For example, a concave curve is one with an inverted U shape. One way to easily remember what a concave function looks like is a sad face.
Although the use we have made of the concavity has been in relation to a curve, the truth is that it is also applicable to mathematical functions and polygons, as we will see later.
How to know if a function is concave?
If the second derivative of a function is less than zero at a point, then the function is concave at that point.
The above can be expressed as follows:
f »(x) <0
For example, we have the function f (x) = -x 2 + 2x + 5. Its first derivative is f ‘(x) = -2x +2 and its second derivative would be f »(x) = -2. Therefore, the function f (x) = x 2 + x + 3 is concave for every value of x, as we see in the graph below, which is a parabola:
Now let’s imagine this other function f (x) = x 3 -5x 2 +7. Its first derivative f ‘(x) = 3x 2 -10x and its second derivative f »(x) = 6x -10. Once we have the second derivative calculated, we must check for what values of x, the function is convex.
So, we set the second derivative equal to 0:
f »(x) = 6x-10 = 0
6x = 10
x = 1.67
Therefore, the function is concave when x is less than 1.67, since the second derivative of the equation is negative. We can check this by replacing different values of x. Likewise, the function is convex when x is greater than 1.67, as we can see in the image below:
Concave polygon
A concave polygon is one where, to join two of its points, a straight line must be drawn that is outside the figure (an exterior diagonal). Also, at least one of its interior angles is greater than 180º. This is the case of, for example, a concave quadrilateral like the one we see below:
The opposite of a concave polygon is a convex one. This is the one where all the interior angles are less than 180º and, to join any two points in the figure, a straight line can be drawn that remains within the polygon.