What are the local extrema of #f(x) = 2 x + 3 /x#?

1 Answer
Nov 28, 2015

Answer:

The local extrema are #-2sqrt(6)# at #x = -sqrt(3/2)#
and #2sqrt(6)# at #x = sqrt(3/2)#

Explanation:

Local extrema are located at points where the first derivative of a function evaluate to #0#. Thus, to find them, we will first find the derivative #f'(x)# and then solve for #f'(x) = 0#.

#f'(x) = d/dx(2x+3/x) = (d/dx2x) + d/dx(3/x) = 2 - 3/x^2#

Next, solving for #f'(x) = 0#

#2-3/x^2 = 0#

#=> x^2 = 3/2#

#=> x = +-sqrt(3/2)#

Thus, evaluating the original function at those points, we get

#-2sqrt(6)# as a local maximum at #x = -sqrt(3/2)#
and
#2sqrt(6)# as a local minimum at #x = sqrt(3/2)#