How do you find the critical numbers for #y = x/(x^2 + 25)# to determine the maximum and minimum?

1 Answer
Jul 7, 2017

By definition the critical numbers of a function are the values of #x# for which:

#f'(x) = 0#

For #f(x) = x/(x^2+25)# we have that:

#f'(x) = ((x^2+25)d/dx x- x(d/dx (x^2+25)))/(x^2+25)^2 = (x^2+25-2x^2)/(x^2+25)^2#

#f'(x) = -(x^2-25)/(x^2+25)^2 #

As the denominator is always positive:

#f'(x) = 0 => (x^2-25) = 0#

So the critical points are #x=+-5# and we can see that:

#f'(x) < 0# for #abs x > 5# and

#f'(x) > 0# for #abs x < 5#

which means that #f(x)# is decreasing in the intervals #(-oo, -5)# and #(5, +oo)# and increasing in the interval #(-5,5)#.

Thus #x=-5# is a local minimum and #x=5# is a local maximum.

graph{x/(x^2+25) [-10, 10, -0.15, 0.15]}