Derivative Calculation: f'(x) & Increase/Decrease

[headbang]

Have a function
\(\displaystyle f(x)=4x^3-x^4\)

Found the x values are X -1, 0, 1, 2, 3 , 4,
f(Y) -5, 0, 3, 16, 27, 0

i Need to find \(\displaystyle f^{\prime}(x)\) and find where it incteases and decreases??\(\displaystyle f`(x)= 3*4x^2-4x^3=4x^2(3-x)\)

what to Next?

 

[Ackbach]

Typically, you use $f'(x)$ to find out where $f(x)$ increases and decreases, not where $f'(x)$ increases or decreases (they're very different things).

What I would do is find out where your derivative changes sign (candidates are the roots of the derivative). I map out the procedure like this:

1. Find out where the derivative is zero. Suppose this happens at $x=-1, 4, 9$.
2. Divide up the whole real line into pieces, depending on where the derivative is zero. In the example, we'd have four pieces: $(-\infty,-1), (-1, 4), (4, 9),$ and $(9,\infty)$.
3. For each interval generated in Step 2, sample the derivative once inside the interval. You only need to do this once, assuming you found your roots correctly. So, let's say we do $f'(-1), f'(0), f'(5),$ and $f'(10)$.
4. For each interval you sampled where the derivative is positive, you have an increasing function. For each interval you sampled where the derivative is negative, you have a decreasing function.

Does that make sense?

 

[headbang]

You are right that i have to use \(\displaystyle f^{\prime}(x)\) to find out where \(\displaystyle f\) increases and decreases. I still don't understand how to calculate this, to solve the problem.

 

[Ackbach]

You computed $f'(x)$ correctly! Go ahead and plug numbers into that derivative.

 

[MarkFL]

You are right that i have to use \(\displaystyle f^{\prime}(x)\) to find out where \(\displaystyle f\) increases and decreases. I still don't understand how to calculate this, to solve the problem.

Follow step 1 of the procedure given by Ackbach above...find the critical numbers, that is, the roots of the derivative. These are the places at which the derivative may change sign. Note that roots of even multiplicity will not be such places. What do you conclude so far?