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 36x2bx.
These curves represent the graphs of the functions f( ; )a b :xax36x2bx.
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 x0?
4. In this question, we will admit that C( ; )a b has a stationary inflexion point at x1.
a) What can we say about f ’(a; b)(1) and f ”(a; b)(1)? b) Deduce the values of a and b.
