The concavity is the property with which a surface or a geometric figure has, so that its central part is deeper than the ends.
We can relate the concavity with the interior of a sphere that has been cut in two, or with the image of a well, since this is a sinking and, therefore, it is concave.
Concavity is the opposite of convexity, which is when a surface or object presents a prominence, as in the case of a mountain, or when we hit ourselves and it causes inflammation.
Of course, depending on the document we read or the explanation, it could be considered that the mountain is concave. It all depends from the point of view you look at.
Concavity of a parable
The concavity of a parabola occurs when it has an inverted U shape, as we can see in the following image.
Now, we know that this parabola is concave because of its function f (x) = -x 2 + 2x + 5, the first derivative is f ‘(x) = -2x + 2, and its second derivative would be f »(x) = -2. Therefore, the function f (x) = x 2 + 2x + 5 is concave for every value of x.
This is true because, to determine the concavity of a function, the second derivative must be evaluated. If this is less than zero, then the function is concave at that specific point or range.
Concavity of a function
As we mentioned previously, if the second derivative of a function is less than 0 in an interval of the domain (set of values that x can take), then it is concave in that interval.
For example, a function can be concave between [4,9] and convex in the interval [10,16].
Now let’s look at a case. If we have the function f (x) = -x 3 + 6x + 5, its first derivative is f ‘(x) = -3x 2 +6 and its second derivative would be f »(x) = -6x. Therefore, the function is concave for every value of x greater than 0 (see image below where the graph becomes an inverted U from positive values of x). For example, when x = 3, the second derivative is negative f »(x) = -3 × 6 = -18.
Concavity of a polygon
When we classify polygons, those with concavity or concaves are those that have at least one of their internal angles greater than 180º. Furthermore, at least one of the diagonals is outside the figure. That is, to join at least two points in the figure, a straight line must be drawn outside the polygon.
For example, let’s look at the following concave trapezoid:
Likewise, the following figure called a star:
See more in concave polygon.