Convex function: senior math course in PDF.

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1 I. Convexity of a function
2 II. Convex function and first and second derivatives
3 III. Tangent and inflection point

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

2. convexity and concavity.

Definitions:

Let f be a function and C_f 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.

Ownership:

The function $x\,\mapsto \,\sqrt{x}$ is concave.

The functions $x\,\mapsto \,x^2$ and $x\,\mapsto \,e^x$ are convex.
The function $x\,\mapsto \,\frac{1}{x}$ is convex on $\mathbb{R}$ .

Example:
Let f be the inverse function defined on $\mathbb{R}^*$ by $f(x)=\frac{1}{x}$ 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 $\mathbb{R}^{+*}$ and the segment [AB] is below the curve for x strictly negative
so f is concave on $\mathbb{R}^{-*}$ .

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 $t\in%5B0;1%5D$ , we have :
$f(tx+\,(1\,-\,t)y)\,\leq\,\,tf(x)+\,(1\,-\,t)f(y)$
– If f is a concave function on an interval I then for all real x and y of I and for all t $t\in%5B0;1%5D$ , we have :
$f(tx+\,(1\,-\,t)y)\,\geq\,\,tf(x)+\,(1\,-\,t)f(y)$

Demonstration:
Let be two real x and y and let be $t\in%5B0;1%5D$ .

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 $f(tx+(1-t)y)\leq\,\,tf(x)+(1-t)f(y)$ .

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 $-\,f$ is concave.

Example:

Let fla function be defined on $\mathbb{R}$ by .

The function $x\,\mapsto \,e^x$ is convex, so $f:x\,\mapsto \,-e^x$ 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 $\mathbb{R}$ .
The table of variations of the function has been drawn up.

Then f is concave on $%5D-\infty\,;\,3%5D$ and convex on $%5B3;+\infty\,%5B$ .

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, f'' , is the derivative of .

Example:
Let f be the function defined (and derivable twice) on $\mathbb{R}$ 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.

Theorem:

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.

Ownership:

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

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

Proof:

Let $\phi$ be the function defined on I by the difference between the function and its tangent.
$\phi(x)=f(x)-(f'(x_0)(x-x_0)+f(x_0))=f(x)-f'(x_0)x+f'(x_0)x_0-f(x_0)$ .
Then $\phi$ is derivable as a sum of derivable functions and, noting $\phi'$ as its derivative, we get :

$\phi'(x)=f'(x)-f'(x_0)+0-0=f'(x)-f'(x_0)$ .

Now is positive so is increasing. Hence:
if $x\geq\,\,x_0$ then $f'(x)\geq\,\,f'(x_0)$ then $\phi'(x)\geq\,\,0$ .
if $x\leq\,\,x_0$ then $f'(x)\leq\,\,f'(x_0)$ then $\phi'(x)\leq\,\,0$ .
In addition, $\phi(x_0)\,=\,f(x_0)-f'(x_0)x_0\,+\,f'(x_0)x_0\,-f(x_0)=0$

We obtain the table of variations below.

Therefore, for any real x of I, $\phi(x)\geq\,\,0$ so $f(x)\geq\,\,f'(x_0)(x-x_0)+f(x_0)$ 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.

Definition:

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

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

Then the origin of the reference frame $O(0\,;\,0)$ 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, f'' must change sign and therefore must change variation.

Example:

If then and .
So $f''(x)\geq\,\,0\Leftrightarrow\,x\geq\,\,0$ and $f''(x)\leq\,\,0\Leftrightarrow\,x\leq\,\,0$ .

There is a change of sign of the second derivative, so f changes convexity, so there is in $O(0\,;\,0)$ an inflection point.

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Convex function: senior math course download PDF.

I. Convexity of a function

II. Convex function and first and second derivatives

III. Tangent and inflection point

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