Thread: Finding the critical numbers

Find critical numbers for x^(4/5) * (x-1)^2?

If you could help out, please. I've already done most of the work, but I'm not too sure on where to go and how to work with the problem to get the critical numbers.

(1). The derivative of the problem. Using the product rule you get

x^(4/5) * 2(x-1) + (x-1)^2 * 4/5x^-1/5

here's where I'm stuck.. the factoring

Relating to the solutions manual for a similar problem ( x^(4/5) * (x-4)^2 ), I know that I'll be able to factor it by:

1/5x^-1/5 * (x-1)

This is where I'm stuck... factoring will I get:

(1/5)x^-(1/5)*(x-1) [ 5 * x * 1 + (x-1) *4] ????

can anyone give me a push on this? I should be able to solve this step after the factoring.

-Thanks

Hello, endiv!

There's some tricky algebra in this one . . .

Find critical numbers for: .$\displaystyle f(x) \:=\:x^{\frac{4}{5}}(x-1)^2$

Differentiate and equal to zero . . .

. . $\displaystyle f\,'(x) \;=\;x^{\frac{4}{5}}\cdot2(x-1) + \tfrac{4}{5}x^{-\frac{1}{5}}(x-1)^2 \;=\;0$

We have: .$\displaystyle 2x^{\frac{4}{5}}(x-1) + \frac{4(x-1)^2}{5x^{\frac{1}{5}}} \;=\;0$

Multiply by $\displaystyle 5x^{\frac{1}{5}}\!:\;\;10x(x-1) + 4(x-1)^2 \;=\;0$

Factor: .$\displaystyle 2(x-1)\bigg[5x+2(x-1)\bigg] \;=\;0 \quad\Rightarrow\quad 2(x-1)(7x-2) \;=\;0$


Therefore, the critical values are at: .$\displaystyle x \;=\;1,\:\tfrac{2}{7}$

Originally Posted by Soroban

Hello, endiv!

There's some tricky algebra in this one . . .

Differentiate and equal to zero . . .

. . $\displaystyle f\,'(x) \;=\;x^{\frac{4}{5}}\cdot2(x-1) + \tfrac{4}{5}x^{-\frac{1}{5}}(x-1)^2 \;=\;0$

We have: .$\displaystyle 2x^{\frac{4}{5}}(x-1) + \frac{4(x-1)^2}{5x^{\frac{1}{5}}} \;=\;0$

Multiply by $\displaystyle 5x^{\frac{1}{5}}\!:\;\;10x(x-1) + 4(x-1)^2 \;=\;0$

Factor: .$\displaystyle 2(x-1)\bigg[5x+2(x-1)\bigg] \;=\;0 \quad\Rightarrow\quad 2(x-1)(7x-2) \;=\;0$


Therefore, the critical values are at: .$\displaystyle x \;=\;1,\:\tfrac{2}{7}$

holy.. wow... I didn't think of it that way! You even did the extra of giving the critical numbers. (embarassed)

Thanks so much!