In mathematics, a concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.
Definition. Formally, a real-valued function f defined on an interval [a,b] is called concave if, for any two points x and y in its domain and any t in [0,1], we have:
This definition merely states that for every z between x and y, the point (z, f(z)) on the graph of f is above the straight line joining the points (x, f(x)) and (y, f(y)).
Suppose that f is differentiable two times on an interval [a,b].
Theorem. (Concavity Test). If f”’> (x) > 0 for all x on an interval [a,b], then the graph of f is concave upward on [a,b]. If f”> (x) < 0 for all x in [ a,b], then the graph of f is concave downward on [a,b].
To graph a function f, follow these steps:
Exercise 1. Take the cubic function and determine its graph.
Solution. (Step 1) The function f(x) is defined for all real numbers. Hence, its domain is: ∀x ∈ ] – ∞, + ∞ [. Now, let’s ﬁnd the zeros, if it is not complicated. We can at least factor out one power of x:
so f(0) = 0. The other factor is a quadratic, so the other two roots are .
We can skip this step for now since the roots are so complicated.
(Step 2). Monotonicity.
To apply the first derivative test for extrema, we solve the following inequality:
and evaluate the sign of f’
Then, applying the first derivative test for extrema, we say:
Moreover, we study the behavior of the function f at the end points of its domain ]- ∞, + ∞ [:
Showing that the horizontal and oblique asymptotes for the graph of f do not exist.
(Step 4.) Assemble monotonicity data
(Step 3). Concavity.
To apply the first concavity test for inflections, we solve the following inequality:
and evaluate the sign of f”.
(Step 4.) Assemble concavity data
Exercise 2. Let the quartic function and determine its graph.
10. Curve Sketching