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: **

- Find when f is positive, negative, zero, not deﬁned.
- Find f’ and form its sign chart. Conclude information about increasing/decreasing and local max/min.
- Find f”(if it is possible!) and form its sign chart. Conclude concave up/concave down and inﬂection.
- Put together a big chart to assemble monotonicity and concavity data.
- Graph the assigned function.

**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.

*1*. Functions and their graph. One–one and onto functions. Composite and inverse functions.

*2*. Linear inequalities. Systems of linear inequalities

*3*. Polynomial inequalities. Rational inequalities. Absolute-value inequalities

*6*. Differentiation. Derivative rules, the chain rule.

*7*. Application of differentiation: L'Hospital's Rule

*8*. Vertical, Horizontal and Slant asymptotes

*9*. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function

*10*. Curve Sketching

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