# Functions Exercise 3 | Logarithmic and Exponential Functions

f is the function defined on ℝ by:
(Cf) 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+
f(x)

and prove that
lim  xf(x)=+

2- a-Prove that for every real number x:
f(x)=12e2x(ex2)(4ex1)
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 (Cf) at +∞ .
b- Analyze the position of (Cf) with respect to (△).
4- Write an equation for (T) tangent to (Cf) at the point with abscissa 0.
5- Construct  (△) and (T) and the curve (Cf) 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)=12e2x+92ex+2x2
(Ch) 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)= af(x)+b
b- Explain how (Ch) can be created based on (Cf) (it does not require creating (Ch)).