What are the absolute extrema of #f(x)=9x^(1/3)-3x in[0,5]#?

1 Answer
Jun 4, 2018

Answer:

The absolute maximum of #f(x)# is #f(1)=6# and the absolute minimum is #f(0)=0#.

Explanation:

To find the absolute extrema of a function, we need to find its critical points. These are the points of a function where its derivative is either zero or does not exist.

The derivative of the function is #f'(x)=3x^(-2/3)-3#. This function (the derivative) exists everywhere. Let's find where it is zero:

#0=3x^(-2/3)-3rarr3=3x^(-2/3)rarrx^(-2/3)=1rarrx=1#

We also have to consider the endpoints of the function when looking for absolute extrema: so the three possibilities for extrema are #f(1), f(0)# and # f(5)#. Calculating these, we find that #f(1)=6, f(0)=0,# and #f(5)=9root(3)(5)-15~~0.3#, so #f(0)=0# is the minimum and #f(1)=6# is the max.