Tell whether the function is increasing or decreasing

Question

Consider the function f(x)=5x+6x^-1. For this function there are four important intervals: (-infinity,A], [A,B), (B,C), and [C,infinity) where A and C are the critical numbers and the function is not defined by B.

Find A:

Find B:

Find C:

For each of the following intervals, tell whether f(x) is increasing or decreasing.

(-infinity, A]

[A,B)

(B,C]

[C,infinity)

 

Answer 1

Checking I found that the derivative of

f(x)=5x+6x^(-1) is equal to

5 - 6x^(-2); if I make it equal to zero to find the inflection points I will get:

x=√(6/5) positive or negative and then I will have to test for positive or negative slope

Answer 2

To find critical numbers find the derivative and set that to so. In this case that's 5-6x^2 so the critical points are plus and minus square root of 5/6. To find where it is decreasing test points on the derivative like negative on, zero and one. At negative one derivative is negative so f is decreasing from negative infinity to that firstcritical poInt. At zero f is increasing. At one f is decreasinG and continues that way to infinity. 

 

As for not defined, we can see that negative exponents mean f is not defined at zero even if the function is increasing. To check plug in plus and minus one half to get a positive slope left and right of zero.