Answer to Question #1430 in Calculus for Lauren

Question #1430
For y= 3x/2e^(x) + e^(-x) use graphing techniques to find the approximate intervals on which the function is <br> A)Increasing<br> B)Decreasing<br> C)concave up<br> D)concave down<br> E) find local extreme values<br> F)Find inflection points<br>
Expert's answer
3x/2ex + e-x = e-x/2 *(3x + 2)

The first derivation of the function is
y&#039; = 3e-x/2 - e-x/2 *(3x + 2) = e-x/2*(3 -3x - 2) = e-x/2 *(1 - 3x)
y&#039; = 0: x = 1/3
The second derivation of the function is:
y&#039;&#039; = -3e-x/2 - e-x(1 - 3x)/2 = e-x/2 *(-3 - 1 + 3x) = e-x/2 *(- 4 + 3x)
y&#039;&#039; = 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.

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