The definition of the **concavity** of a graph is introduced along with inflection points. Examples, with detailed solutions, are used to clarify the concept of concavity.

## Example 1: Concavity UpLet us consider the graph below. Note that the slope of the tangent line (first derivative) increases. The graph in the figure below is calledconcave up.
Figure 1
## Example 2: Concavity DownThe slope of the tangent line (first derivative) decreases in the graph below. We call the graph belowconcave down.
Figure 2
## Definition of ConcavityLet f ' be the first derivative of function f that is differentiable on a given interval I, the graph of f is(i) concave up on the interval I, if f ' is increasing on I
, or (ii) concave down on the interval I, if f ' is decreasing on I.
The sign of the second derivative informs us when is f ' increasing or decreasing.
## TheoremLet f '' be the second derivative of function f on a given interval I, the graph of f is(i) concave up on I if f ''(x) > 0 on the interval I.
(ii) concave down on I if f ''(x) < 0 on the interval I.
## Example 3Determine the values of parameter a for which the graph of function f, defined below, is concave up or down.f(x) = a x^{ 2} + b x + c## Solution to Example 3We first find the first an second derivatives of function f.f '(x) = 2 a x + b f ''(x) = 2 a We now study the sign of f ''(x) which is equal to 2 a. If a is positive, f ''(x) is positive in the interval (-∞ , + ∞). According to the theorem above, the graph of f will be concave up for these positive values of a. If a is negative, the graph of f will be concave down on the interval (-∞ , + ∞) since f ''(x) = 2 a is negative. An interactive applet on the concavity of graphs quadratic functions is in this site and you can verify the results of this example.
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