Increasing and decreasing functions

Question

F (x)=(12x-3)e^x

Answer 1

Pat,

 

I believe your problem is to determine the intervals where this function is increasing or decreasing.

 

To do this, find the derivative of f(x). This can be found using the Product Rule and the answer is

 

f′(x) = e^x(12x + 9).

 

Next, find the critical numbers by setting this equal to zero and solving for x. Since e^x is never zero, you will get x = −3/4 as the critical number.

 

Now, you can divide the domain of this function (all real numbers) into two intervals: (−∞,−¾) and (−¾,∞). On each of these, the function will have either a positive or a negative derivative throughout. By choosing a value in each and plugging it into f′(x), you can determine what the sign of f′ is on the entire interval.

 

I chose x = -1 and x = 0 for my test points. Plugging these in, I can see that

 

f′(−1) = −3e⁻¹ < 0, which implies that f is decreasing on the first interval (−∞,−¾)

 

and

 

f′(0) = 9e⁰ = 9 > 0, which implies that f is increasing on the second interval (−¾,∞).

 

I hope this answers your question. You can also see that f has a local minimum at x = −¾, if you understand what that means.