What are the critical values, if any, of #f(x) = x^(7/3) - 7x^(1/3) #?

1 Answer
Jim H
Nov 2, 2015

The critical values are #-1, 0, 1#

Explanation:

For #f(x) = x^(7/3) - 7x^(1/3) #, note that the domain of #f# is #(-oo,oo)#.

#f'(x) = 7/3x^(4/3)-7/3x^(-2/3)#

# = (7x^(4/3))/3-7/(3x^(2/3))#

# = (7x^2-7)/(3x^(2/3))#

#'(f)# does not exist for #x=0# and #f'(x) = 0# for #x=+-1#.

Because all of these values are in the domain of #f#, all three are critical values for #f#.

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