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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.**

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.**

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

We say:

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

**3. the usual functions.**

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.**

– 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.

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.**

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.**

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:

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

**3. convexity and second derivative.**

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

- f is convex on I if and only if, for any real x of l, is positive.
- 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.**

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 .

In addition,

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:

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

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

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.

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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