How to Find Where Functions INCREASE and DECREASE

[Lime]

f(x) = (x - 2)e^x

How would you find out where f is increasing and decreasing?

 

[galactus]

Check your derivatives.

If f'(x)>0, then f is increasing

If f'(x)<0, then f is decreasing.

If f'(x)=0, then f is constant. This may be a point where there is a transition from decrease to increase or vice versa. If f'(x)=0 at a single point and not over an interval, then it is a transition.

 

[Lime]

The answer book says the derivative of the original function is

f'(x) = e^x + (x - 2)e^x = e^x(x - 1)

How is this so? The derivative of (x - 2) is 1. Multiply that by the derivative of e^x, e^x, and the derivative of the original function should be e^x. No?

 

[galactus]

That's correct...No

It doesn't work that way. Use the product rule. The derivative of a product is generally not the product of the derivatives

Product rule:

\(\displaystyle (x-2)e^{x}+e^{x}(1)=e^{x}(1+(x-2))=e^{x}(x-1)\)

 

[Lime]

And apparantly you need to set the derivative to equal zero in order to find the x which I think represents the transition.

Anyway, the answer book has x = 1. How do they get that number?

 

[galactus]

Set the derivative to 0 and solve for x. You can easily see if you enter x=1 into

the function you get 0. It's factored.

Here's a graph so you can see where it's concave up and down.

Did you try graphing it?. That helps.