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

Question

Elijah Abram · Accepted Answer

#(-oo, -1/sqrt(2))#
#(-1/sqrt(2), 1/sqrt(2))#
#(1/sqrt(2), oo)# The function is expressed as #y= 2x + x^-1# Which sets itself apart from: #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)#. I hope this is useful!

Scarlett Allison · Answer

undefined To find the intervals of increasing and decreasing using the first derivative for the function y = 2x + 1/x:

1. Find the first derivative of the function y' = d(2x + 1/x)/dx.
2. Set y' equal to zero and solve for x. These points are potential critical points.
3. Determine the sign of the first derivative in intervals separated by the critical points found in step 2.
   - If y' > 0, the function is increasing in that interval.
   - If y' < 0, the function is decreasing in that interval.
4. These intervals where the first derivative is positive represent where the original function is increasing, and where it's negative represent where it's decreasing.