Applications of differentiation: concave and convex functions
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)).
Applications of differentiation: concave and convex functions
Applications of differentiation: concave and convex functions
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].
Graphing Checklist
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.
Exercises
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.
Exercises
(Step 2). Monotonicity.
To apply the first derivative test for extrema, we solve the following inequality:
and evaluate the sign of f’
![\text{The function}\,\,f\,\,\text{is decreasing on}\,\,]-1,2 [\,\,](http://podcast.federica.unina.it/files/formule/tex_632c459017c483ec895530111b940f84.gif)
![\text{and is increasing on}\,\,]-\infty,-1[\cup]2,+\infty[](http://podcast.federica.unina.it/files/formule/tex_a4525c6430a8c85168e82e5cbc735817.gif)
Exercises
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
Exercises
(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
![\text{The function}\,\,f\,\,\text{is concave down on}\,\,]-\infty,1/2[\,\,<br />](http://podcast.federica.unina.it/files/formule/tex_44ef874a20f6b0552528f033d2bbf382.gif)
![\text{and is concave up on}\,\,]1/2,+\infty[](http://podcast.federica.unina.it/files/formule/tex_2a4192a6d0cf33dd2266bf0c083f1377.gif)
Exercises
(Step 5). Graph the function.
Exercises
Exercise 2. Let the quartic function 
and determine its graph.
Le lezioni del Corso
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
