Find the critical numbers of f(x)= 9x^(2/3) - 3x

Critical values occur when the first derivative is zero and when the first derivative is undefined.

 

f(x)  =  9x^2/3 - 3x     --->     f'(x)  =  6x^-1/3 - 3

 

f'(x) is undefined when x = 0; so one critical value is 0.

 

f'(x) = 0     --->     6x^-1/3 - 3  =  0     --->     6x^-1/3  =  3     --->     x^-1/3  = 1/2     --->     x^1/3  =  2     --->   x  =  8

 

Thus, another critical value is 8.