Question

how to fine critical numbers of f, describe the open intervals on which f is increasing or decreasing, and locate all relative extrema.

Answer 1

x^(2/3)(x-5)

Answer 2

i found the critical point, x=2 but i dont know what else to do.

Answer 3

First, I need some clarification:

By x^(2/3)(x-5), do you mean the following?

\[y=\left( x ^{2/3} \right)\left( x-5 \right)\]

Answer 4

yes

Answer 5

A critical point is when f'(x) = 0 or f'(x) is undefined.
The x = 2 you found is when f'(2) = 0
But f'(x) is undefined at x = 0 and therefore
x = 0 and x = 2 are critical points.

Split the domain into (-infinity, 0) ; (0, 2) and (2, infinity) and analyze the behavior of the function by examining f'(x) in these three intervals.

Answer 6

would those 3 intervals be open intervals? or wat

Answer 7

Yes. We are only interested in the sign of f'(x) in those three intervals. It is open interval because we already know that f'(0) is undefined and f'(2) = 0 and so we don't need to include them in the interval.

Answer 8

Building on what ranga has said ("Split the domain into (-infinity, 0) ; (0, 2) and (2, infinity) and analyze the behavior of the function by examining f'(x) in these three intervals."):

Choose a test number from each interval. For example, you might choose the test number -1 from the first interval, 1 from the second, and 3 from the third.

Evaluate the sign of the first derivative f'(x) at each of these test numbers.

If, on one of the intervals, f'(x) is positive, then the function f(x) is INCREASING on that interval.
If, on one of the intervals, f'(x) is negative, then the function is DECREASING on that interval.

This is what ranga meant in recommending that you"analyze the behavior of the function on these three intervals." Good luck!