How do you find the intervals of increasing and decreasing using the first derivative given #y=2x+1/x#?

1 Answer
Noah G
May 27, 2017

#(-oo, -1/sqrt(2))#
#(-1/sqrt(2), 1/sqrt(2))#
#(1/sqrt(2), oo)#

Explanation:

The function can be written as

#y= 2x + x^-1#

Which differentiates to:

#y' = 2 - x^-2 = 2 - 1/x^2#

A function will change from increasing to decreasing and vice versa when the derivative equals #0#.

#0 = 2 - 1/x^2#

#1/x^2 = 2#

#1/2 = x^2#

#x =+-1/sqrt(2)#

Now select a test point, let it be #x = 1#. Evaluating within the derivative gives a positive value which means the function is increasing at that Point. It also means that the function is decreasing on #-1/sqrt(2) < x < 1/sqrt(2)#.

Hopefully this helps!

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