finding local max of function via 1st derivative

**bcahmel** · December 12th 2010, 07:02 AM

y=x(1-x)^(1/3)

The derivative of the function turns out to be (1-x)^(1/3) - x/(3(1-x)^2/3)
Using the calculator I found out the max is 0.750, .472...But when I set the derivative equal to 0...and tried to factor I got x=-3/11, x=1...I understand how to find the mix and max, my question is how to factor this particular equation...

**adkinsjr** · December 12th 2010, 07:18 AM

When you set the derivative equal to zero, you need to do the following algebra to get the critical point:

$0=(1-x)^{\frac{1}{3}}-\frac{x}{3(1-x)^{\frac{2}{3}}}$

$(1-x)^{\frac{1}{3}}=\frac{x}{3(1-x)^{\frac{2}{3}}}$

$1-x=\frac{x^3}{27(1-x)^2}$

$27(1-x)^3=x^3$

$3(1-x)=x$

$3=4x$

$x=.75$

**chisigma** · December 12th 2010, 07:24 AM

Originally Posted by bcahmel

y=x(1-x)^(1/3)

The derivative of the function turns out to be (1-x)^(1/3) - x/(3(1-x)^2/3)
Using the calculator I found out the max is 0.750, .472...But when I set the derivative equal to 0...and tried to factor I got x=-3/11, x=1...I understand how to find the mix and max, my question is how to factor this particular equation...

Is...

$\displaystyle y^{'}= (1-x)^{\frac{1}{3}} - \frac{x}{3}\ (1-x)^{-\frac{2}{3}} = (1-x)^{\frac{1}{3}}\ \{1 - \frac{x}{3\ (1-x)}}\}$ (1)

... so that the maximim is for $x=\frac{3}{4}$ ...

Merry Christmas from Italy

$\chi$ $\sigma$

**Soroban** · December 12th 2010, 07:34 AM

Hello, bcahmel!

$y\:=\:x(1-x)^{\frac{1}{3}}$

$\text{The derivative turns out to be: }\:y' \:=\: (1-x)^{\frac{1}{3}} - \dfrac{x}{3(1-x)^{\frac{2}{3}}}$

$\text{Using the calculator I found out the max is: }\:(0.75,\;0.47247\hdots)$

We have: . $(1-x)^{\frac{1}{3}} - \dfrac{x}{3(1-x)^{\frac{2}{3}}} \;=\;0$

Assuming $x \ne 1$ , multiply through by $3(1-x)^{\frac{2}{3}}\!:$

. . $3(1-x) - x \;=\;0 \quad\Rightarrow\quad 3 - 3x - x \:=\:0$

. . $-4x \:=\:-3 \quad\Rightarrow\quad x \:=\:\dfrac{3}{4}$

**bcahmel** · December 13th 2010, 12:17 PM

thanks to everyone for the help! I appreciate it greatly

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