Answer to Question #1430 in Calculus for Lauren

Question

For y= 3x/2e^(x) + e^(-x) use graphing techniques to find the approximate intervals on which the function is

A)Increasing

B)Decreasing

C)concave up

D)concave down

E) find local extreme values

F)Find inflection points

Answer 1

3x/2e^x + e^-x = e^-x/2 *(3x + 2)

The first derivation of the function is
y' = 3e^-x/2 - e^-x/2 *(3x + 2) = e^-x/2*(3 -3x - 2) = e^-x/2 *(1 - 3x)
y' = 0: x = 1/3
The second derivation of the function is:
y'' = -3e^-x/2 - e^-x(1 - 3x)/2 = e^-x/2 *(-3 - 1 + 3x) = e^-x/2 *(- 4 + 3x)
y'' = 0: x = 3/4
The fustion increases on the (-inf, 1/3) and decreases on (1/3, inf)
The local extreme is x= 1/3;
The fuction concaves up on (-inf, 3/4), concaves down on (3/4, inf)
The inflection point is 3/4.