d2 1 differentiation

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BACCALAURÉAT GÉNÉRAL ET TECHNOLOGIQUE SESSION 2010

ÉPREUVE SPÉCIFIQUE MENTION « SECTION EUROPÉENNE OU DE LANGUE ORIENTALE » Académies de Paris-Créteil-Versailles

Binôme : Anglais / Mathématiques DIFFERENTIATION

Sujet D2 - 1

The first part of this page is a summary that can be helpful to do the exercise.

If a curve y f x( ) has a tangent line Tx₀ at x x 0, the slope of Tx0 (or gradient of the curve at x0) is

given by 0 0 0 0 ( ) ( ) '( ) lime f x e f x f x e   

 . This defines the first order derivative function f ' of the

function f .

Similarly, the second order derivative function, f '' (f double dash), of the function f is defined to be the first order derivative of the first order derivative of the function f .

A curve y f x( ) has a stationary point at x x 0 means that f x'( ) 00  .

Stationary points are of three types:

 local maximum point if when passing through the point, the gradient of the curve changes from a

negative to a positive sign.

 local minimum point if when passing through the point, the gradient of the curve changes from a

positive to a negative sign.

 stationary inflexion point if when passing through the point, the gradient of the curve keeps the

same sign.

If the curve y f x( ) has a stationary inflexion point at x x 0, then f ”(x0) = 0.

If ( ) n

f x a x , with a a given real number, then '( ) n 1

f x nax  .

Exercise:

In this exercise, we study the family of cubic graphs C( ; )a b : y ax 36x2bx.

These curves represent the graphs of the functions f( ; )a b :xax36x2bx.

1. Quickly explain why all these curves C( ; )a b pass through the origin (0;0).

Then express, in terms of a and b, the slope of the tangent line to C( ; )a b at (0;0).

2. a) Find out the x - intercept and the y- intercept of C( 2; 6) . b) Sketch the curve C( 2; 6) .

3. Is it possible to find values for a and b so that C( ; )a b has a stationary inflexion point at x0?

4. In this question, we will admit that C( ; )a b has a stationary inflexion point at x1.

a) What can we say about f ’(a; b)(1) and f ”(a; b)(1)? b) Deduce the values of a and b.