Convexity and concavity. Point of inflection

Let be a function differentiable on an interval J. The function f is called convex (concave) on J, if all points of its graph on J lie above (below) any tangent line to on this interval (excepting point of tangency). Let f be continuous at a point . If there exists such that f is concave (convex) in and convex (concave) in , the point is called the point of inflection of .

The second Derivative Test for Concavity and Convexity:

If , for each , then is convex on J,

if , for each , then is concave on J.

It follows:

If is continuous at and () in and () in , then is a point of inflection.

Moreover: If is a point of inflection of f, then either or doesn’t exist.

If f is three times differentiable at a point , and , then is a point of inflection.

Definition 6. Given that is continuous on , if any such that

(i) (ii)

Concave Downward Concave Upward

 

Theorem 3. If is a function on such that is second differentiable on then

(i) iff is concave upward on

(ii) iff is concave downward on .

 

Definition 7. Let be a continuous function. A point on the graph of is a point of inflexion (point of inflection) if the graph on one side of this point is concave downward and concave upward on the other side. That is, the graph changes concavity at .

Note A point of inflexion of a curve must be a continuous point but need not be differentiable there. In Figure (c), R is a point of inflexion of the curve but the function is not differentiable at .

Theorem 4. If is second differentiable function and attains a point of inflexion at , then

.

Note:

(i) max. or min. point but not derivative.

(ii) point of inflexion may not be obtained by solving where and such that .

(iii) Let be a function which is second differentiable in a neighborhood of a point of inflexion iff does not change sign as increases through (sign gradient test)

– if and , then attains a relative max. or relative min.

– if and , then attains an inflexion point at .