critical numbers of x^(7/9) [10 - x] (can't get answer)

[cbaechle]

I have a practice problem from a quiz generated by Maple TA (Maple TA will show you the answer and the work involved, but many times the process is overly convoluted).

I'm being asked to find the critical numbers of the following:
\(\displaystyle x^\frac{7}{9}(10-x)\)
The answer given is \(\displaystyle 0 ; \frac{35}{8}\)

I have no clue how they got it! My first and only step I've figured out so far is to get the first derivative.

\(\displaystyle (x^\frac{7}{9})(-1)+(10-x)(\frac{7}{9}x^\frac{-2}{9})\)

I obviously need to reduce and change form somehow. Maple TA gives a little help. It shows the derivative as
\(\displaystyle \frac{1}{9} \frac {70-16x}{x ^ \frac{2}{9}}\)

From there it's easy to see why you get the answers, but I have no clue how to simplify my derivative to get to that point.

 

[galactus]

Re: critical numbers

If we diff. we get:

\(\displaystyle =\frac{70}{9}x^{\frac{-2}{9}}-\frac{16}{9}x^{\frac{7}{9}}\)

Factor out \(\displaystyle \frac{2}{9}x^{\frac{-2}{9}}\):

\(\displaystyle \frac{2(35-8x)}{9x^{\frac{2}{9}}}=0\)

\(\displaystyle 8x-35=0\)

\(\displaystyle x=\frac{35}{8}\)

It can easily be seen that 0 is the other.