# Thread: Basic differentiation - find increasing and decreasing intervals...

1. ## Basic differentiation - find increasing and decreasing intervals...

I have the following question to solve but I don't seem to grasp the concept very clearly (excuse me, I am self-studying).

For each of the following functions f(x), find f'(x), the intervals in which f(x) is decreasing, and the intervals in which f(x) is increasing.

x^3/2 ( x - 1 ) for x > 0

I cannot seem to get the derivative function f'(x) right here.

This is what I reached:

=> f(x) = x^5/2 - x ^ 3/2
=> f'(x) = 5/2x^3/2 - 3/2x^(3/2 - 1)
= 5/2x^3/2 - 3/2x^1/2
= 5/2x^3/2 - 3/2x^1/2

But I don't think it's right because it doesn't help me progress to find the interval where it's decreasing or increasing.

Edit:

I have now grouped it as the following:

1/2x ^ 1/2 (5x - 3)

From my knowledge, I have to solve the inequality 5x - 3 <= 0 and 5x - 3 >= 0 .. But that's not working.

2. Originally Posted by struck
I have the following question to solve but I don't seem to grasp the concept very clearly (excuse me, I am self-studying).

For each of the following functions f(x), find f'(x), the intervals in which f(x) is decreasing, and the intervals in which f(x) is increasing.

x^3/2 ( x - 1 ) for x > 0 Keep this constraint in mind!

I cannot seem to get the derivative function f'(x) right here.

This is what I reached:

=> f(x) = x^5/2 - x ^ 3/2
=> f'(x) = 5/2x^3/2 - 3/2x^(3/2 - 1)
= 5/2x^3/2 - 3/2x^1/2
= 5/2x^3/2 - 3/2x^1/2

But I don't think it's right because it doesn't help me progress to find the interval where it's decreasing or increasing.

Edit:

I have now grouped it as the following:

1/2x ^ 1/2 (5x - 3)

From my knowledge, I have to solve the inequality 5x - 3 <= 0 and 5x - 3 >= 0 .. But that's not working.
All your considerations and all your calculations are correct. You only have to add to your inequalities the condition $\displaystyle x > 0$

Then you'll get:

$\displaystyle f(x)\ is\ \left\{\begin{array}{l}decreasing\ if\ 0<x\leq\dfrac35 \\ increasing\ if\ x > \dfrac35 \end{array}\right.$

I've attached the graph of the function.