How do you find the critical numbers of # f(x)=x^(1/5)-x^(-4/5)#?

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Answer 1 · 2017-07-03T18:33:20

The only critical number is #x=-4#

The critical numbers of a function are the points where its derivative equals zero, which means they are the solutions of the equation:

#(df)/dx = 0#

#1/5x^(-4/5)+4/5x^(-9/5) = 0#

#1/5x^(-4/5)(1+4/x) = 0#

As #x^(-4/5) !=0# for any value of #x# the only solutions are given by:

#1+4/x = 0#

#4/x = -1#

#x=-4#