Monotonicity problem

• Mar 29th 2011, 06:28 PM
steph3824
Monotonicity problem
Establish the inequality e^(x)<= 1/(1-x) for all x<1. A note in my book says to interpret as a monotonicity statement about the function f(x)=(1-x)e^x.

Here is what I have so far:
e^(x)<= 1/(1-x) becomes (1-x)e^x<=1
The derivative of (1-x)e^x is (1-x)e^(x)+(-1)e^x=-xe^x.

Is this all correct so far? I am not sure where to go with this problem now. Help please!
• Mar 29th 2011, 07:05 PM
tonio
Quote:

Originally Posted by steph3824
Establish the inequality e^(x)<= 1/(1-x) for all x<1. A note in my book says to interpret as a monotonicity statement about the function f(x)=(1-x)e^x.

Here is what I have so far:
e^(x)<= 1/(1-x) becomes (1-x)e^x<=1
The derivative of (1-x)e^x is (1-x)e^(x)+(-1)e^x=-xe^x.

Is this all correct so far? I am not sure where to go with this problem now. Help please!

Yes, it is correct...and checking the sign of the derivative you have the answer.

By the way, in the question it should be "for $1\neq x\geq 0$ or something like this...check it.

Tonio