F(X) = (2/3)(X-2)^(-1/3)

F'(X)= (-2/9)(X-2)^(-4/3)

F''(X) = (8/27)(X-2)^(-5/3) Is F''(X) correct???

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- Jul 6th 2009, 06:49 PMBrazucaSimple Prime Question
F(X) = (2/3)(X-2)^(-1/3)

F'(X)= (-2/9)(X-2)^(-4/3)

F''(X) = (8/27)(X-2)^(-5/3) Is F''(X) correct??? - Jul 6th 2009, 06:50 PMRapha
- Jul 6th 2009, 06:55 PMBrazuca
Wasn't a typo. I just had it incorrect.

Thanks now I understand how it works. - Jul 6th 2009, 07:24 PMBrazuca
I have two more questions involving this problem.

F(X) = (2/3)(X-2)^(-1/3) when X = 1 why is the final answer negative?

F'(X)= (-2/9)(X-2)^(-4/3) when X = 1 why is the final answer negative? - Jul 6th 2009, 07:41 PMRapha
Just substitute X=1, then

$\displaystyle F(1) = \frac{2}{3}*(1-2)^{-1/3} = \frac{2}{3}*(-1)^{-1/3}$

We know that $\displaystyle a^{-1} = \frac{1}{a}$

Therefor

$\displaystyle F(1) = \frac{2}{3}*(-1)^{-1/3} = \frac{2}{3}*\frac{1}{\displaystyle(-1)^{1/3}}$

Furthermore we know that $\displaystyle x^{1/2} = \sqrt{x}$ and $\displaystyle x^{1/3} = \sqrt[3]{x}$

Thus

$\displaystyle F(1) = \frac{2}{3}*\frac{1}{\displaystyle(-1)^{\displaystyle1/3}} = \frac{2}{3}*\frac{1}{\sqrt[3]{-1}} = \frac{2}{3}*(-1) = - 2/3 < 0 $

Same here

$\displaystyle F'(1) = (-2/9) (1-2)^{-4/3} = -2/9*(-1)^{\displaystyle-4/3} = -2/9 * \frac{1}{(-1)^{\displaystyle4/3}}$

$\displaystyle = -2/9 * \frac{1}{\displaystyle((-1)^4)^{\displaystyle1/3}} = -2/9*\frac{1}{1^{\displaystyle1/3}} = -2/9 * \frac{1}{\sqrt[3]{1}} = -2/9*1 = -2/9 <0$