Solve the question regarding the four intervals

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 at B.

Find A:

and B:

and C:

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

(-infinity, A]:

[A,B):

(B,C]:

[C,infinity):

 

please provide final answers. 

 

Answer 1

f(x) = 5x + 6/x

 

f(x) is undefined at x = 0, so x=0 is one of the three points.

 

Now let's find the extrema (maximums and minimums).  Take the derivative of f(x), set it to zero, and solve for x:

 

df(x)/dx = 5 - 6/x²

0 = 5 - 6/x²

6 = 5x²

6/5 = x²

 

±(6/5)^1/2 = x

 

So there are extrema at = -(6/5)^1/2 and +(6/5)^1/2.  The three points, then, are:

 

A = -(6/5)^1/2

B =  0

C = +(6/5)^1/2

 

For the specified intervals:

 

As x--> -∞, f(x) --> -∞.  So on the interval (-∞, -(6/5)^1/2], f(x) is increasing to a maximum at f(-(6/5)^1/2)

 

As x --> 0-, f(x) --> -∞.  So on the interval [-(6/5)^1/2,0), f(x) is decreasing from its maximum at f(-(6/5)^1/2) and approaching the asymptote x=0.

 

As x --> 0+, f(x) --> +∞.  So on the interval (0,+(6/5)^1/2]. f(x) is decreasing along the asymptote x=0 to a minimum at f(+(6/5)^1/2)

 

As x --> +∞, f(x) --> +∞.  So on the interval [+(6/5)^1/2,+∞), f(x) is increasing from the minimum at f(+(6/5)^1/2)