Concavity is the property of curving a graph of the function upward or downward. The interval on which derivative of the function, f', increases or decreases can determine where the graph of f is curving upward or curving downward. The first derivative test tells us where a function is increasing or decreasing. And the second derivative gives the information that function is concave upward or concave downward.

Let f be differentiable on an open interval I.
  1. If f' is increasing on I, then the graph of f concave up on I.
  2. If f' is decreasing on I, then the graph of f concave down on I.

Concavity Definition
A point P on the curve $y$ = $f(x)$ is called the inflection point if $f$ is continues there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.

Inflection point: It is a point where a curve changes its direction of concavity. Or in view of concavity we can say that , there is a point of inflection at any point where the second derivative changes sign.
Concavity for any function f(x) shows the shape of graph of function. Graph of any function may be concave up or concave down or some part of graph will be concave up and some part will be concave down. To know whether graph is concave up or concave down we will need to find second derivative of function, f’’(x) if for any given interval of domain [a, b] is positive then graph will be concave up and if f’’(x) for this given interval is negative then graph will be concave down.

The following figure shows the graphs of two increasing function on interval (a,b). Both the graph join point C to D but they look different because they bend in different direction.

In-order to distinguish between these two types of behavior draw tangent to these curve at several point which is shown below.
Concave Up Graph

If the curve lies above the tangents and $f$ is called concave upward on (a, b).

Concave Down Graph

The curve below the tangents and $g$ is called concave downward on (a, b).

If the graph $f$ lies above all the of its tangent on an interval $l$, then it is called concave upward on $l$. if the graph of $f$ lies below all of its tangent on $l$, it is called concave downward on $l$.

Now lets see how the second derivative helps to determine the intervals of continuity. From concave up graph we can see that, going from left to right the slope of the tangent increases. This means that the first derivative is an increasing function therefore the second derivative is positive.

Similarly, from concave down graph we can see that, going from left to right the slope of the tangent decreases. This means that the first derivative is an decreasing function therefore the second derivative is negative.
  1. If the second derivative $f ''(x)$ > $0$ for all $x$ in interval $I$, then the graph is concave upward on $I$.
  2. If the second derivative $f ''(x)$ < $0$ for all $x$ in interval $I$, then the graph is concave downward on $I$.
The following problem will help us to learn more about concavity.

Solved Example

Question: Find inflection points & determine concavity for $f(x)$ = $\frac{x^{5}}{20}$ + $\frac{x^{4}}{12}$ - $\frac{x^{3}}{3}$ + $10$
Solution:
 
We have $f(x)$ = $\frac{x^{5}}{20}$ + $\frac{x^{4}}{12}$ - $\frac{x^{3}}{3}$+$10$

First derivative of the given function is given by

$f'(x)$ = $\frac{x^{4}}{4}$ + $\frac{x^{3}}{3}$ - $x^{2}$

Second derivative of the given function is given by

$f''(x)$ = $x^{3}+x^{2}–2x$ = $x(x^{2}+x-2)$

= $x(x+2)(x-1)$

$f''(x)$ = $0$

$x$ = $0, -2, 1$

Let $-5, -1, 0.5$ be the test points

f ’’(-5) < 0

f’’(-1) > 0

f ’’(.5) < 0

f ’’(10) > 0

Inflection pts: $x$ = $-2, 0, 1$

Curved in up: $(-2, 0),(1, \infty)$

Curved in down: $(-\infty, -2),(0, 1)$