*f*is the function defined on ℝ by:

(C

*f*) is the graph of the function*f*in the coordinate plane associated with the orthogonal and homogeneous vectors (o;➡i,➡j).**1-**Calculate

lim x→+∞

lim x→+∞

*f*(x)and prove that

**lim x→−∞**

*f*(x)=+∞**2- a-**Prove that for every real number x:

*f′(x)=−12e−2x(ex−2)(4ex−1)***b-**Prove that f is strictly decreasing on both intervals ]-∞, -ln4] and [ln2, +∞[ And strictly increasing on [-ln4, ln2] and create a table of its variations.

**3- a-**Show that the line (△) with the equation y=-2x+4 is asymptotic to the curve (C

*f*) at +∞ .

**Analyze the position of (C**

*b-**f*) with respect to (△).

**4-**Write an equation for (T) tangent to (C

*f*) at the point with abscissa 0.

**5-**Construct (△) and (T) and the curve (C

*f*) on the interval [-1.9,+∞] ( We take

*f*(-1.9)≃0 and

*f*(-ln4)=-3.2 ).

6- h iis the function defined on ℝ by:

*h(x)=−12e−2x+92e−x+2x−2*(C

*h*) h its graphical representation in the previous coordinate plane.**a-**Determine the real numbers a and b, where for each real number x,

*h*(x)= a

*f*(x)+b

**b-**Explain how (C

*h*) can be created based on (C

*f*) (it does not require creating (C

*h*)).