The convex function is an essential chapter for the student to understand. This will allow him to progress well in Math.

I. Convexity of a function

1. secant to the representative curve of a function.

Definition:

Let f be a function and its representative curve in a reference frame.
Let A and B be two points of then the line (AB) is secant of .

2. convexity and concavity.

Definitions:

Let f be a function and its representative curve in an orthonormal plane.

We say:

1. f is convex on an interval I if, for any real x of I, is below its secants.
2. f is concave on an interval I if, for any real x of I, is above its secants.

3. the usual functions.

Ownership:

The function is concave.

The functions and are convex.
The function is convex on .

Example:
Let f be the inverse function defined on by and its representative curve
in the mark below.

Then the segment [CD] is above the curve of for x strictly positive so f is
convex on and the segment [AB] is below the curve for x strictly negative
so f is concave on .

4.position with respect to the secants.

Ownership:

– If f is a convex function on an interval I then for all real x and y of I and for all , we have :

– If f is a concave function on an interval I then for all real x and y of I and for all t , we have :

Demonstration:
Let be two real x and y and let be .

Let and ; Then the point
belongs to the segment [AB], secant of .

f being convex, this secant is located above .
M is therefore located above .
Hence .

Remark:

If the previous inequalities are strict, we say that f is a strictly convex or strictly concave function on l.

Property: concavity.

is convex on I if and only if is concave.

Example:

Let fla function be defined on by .

The function is convex, so is concave.

II. Convex function and first and second derivatives

1.convex and concave function.

Theorem:

Let I be a real interval.
Let f be a function twice derivable on I and its derivative function.

• f is convex on l, if and only if, for any real x of l, is increasing.
• f is concave on l, if and only if, for any real x of l, is decreasing.

Example:
Let f be the function defined and derivable on .
The table of variations of the function has been drawn up.

Then f is concave on and convex on .

2. the second derivative function.

Definition:

Let f be a function assumed to be twice derivable on I and its derivative function.
The second derivative of the function f, , is the derivative of .

Example:
Let f be the function defined (and derivable twice) on by the expression
So and .

Remarks:

1. The second derivative of an affine function is always zero.
2. The exponential function is equal to its derivative, so its second derivative as well.

3. convexity and second derivative.

Theorem:

Let f be a function assumed to be twice derivable and its derivative function.

1. f is convex on I if and only if, for any real x of l, is positive.
2. f is concave on I if and only if, for any real x of l, is negative.

Demonstration:

f’ is increasing (resp. decreasing) if and only if is is positive (resp. negative).
Therefore f is convex (resp. concave) if and only if is positive (resp. negative).

III. Tangent and inflection point

1. second derivative and tangent.

Ownership:

Let f be a function assumed to be twice derivable on I with second derivative .

If is positive on I, then the representative curve of f is above its tangents.

Proof:

Let be the function defined on I by the difference between the function and its tangent.
.
Then is derivable as a sum of derivable functions and, noting as its derivative, we get :

.

Now is positive so is increasing. Hence:
if then then .
if then then .

We obtain the table of variations below.

Therefore, for any real x of I, so in other words, the representative curve of f is above its tangents.

Conclusion:

If is positive, then the representative curve of f is above its tangents.

Remarks:

1. If is negative on I then the representative curve of f is below its tangents.
2. Be careful with the reciprocal, a convex function is not necessarily twice derivable.

2.inflection point on the representative curve of a function.

Definition:

Let f be a function twice derivable on an interval I and its representative curve on this interval
in an orthonormal reference frame of the plane.
Let A be a point of and the tangent at point A.
We say that A is an inflection point for if, at point A, the curve passes through .

Example:
Let f be the cube function and its representative curve in a reference frame.

Then the origin of the reference frame is an inflection point for .

On the other hand, the tangents at -1 and 1 do not cross the curve, so the points with coordinates and are not inflection points.

Ownership:

For there to be an inflection point, must change sign and therefore must change variation.

Example:

If then and .
So and .

There is a change of sign of the second derivative, so f changes convexity, so there is in an inflection point.

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