differentiate and find maximum and minimum turning point

**salma2011** · October 1st 2011, 08:42 AM

2(x^2/3) + (x^4/5) +11

i differnetiated this to get
4/3(x^-1/3) + 4/5(x^-1/5)

i know then, that i have to set the differentiated formula = 0. Please help with what i have to do next.

Thanks

**Siron** · October 1st 2011, 08:46 AM

The first derivative is indeed:
$f'(x)=\frac{4}{3}x^{\frac{-1}{3}}+\frac{4}{5}x^{\frac{-1}{5}}$
To find the minumum/maximum then you have to solve $f'(x)=0$ . What do you get? Afterwards make a sign table to determine where the function increases/decreases.

**salma2011** · October 1st 2011, 08:59 AM

I can't remember how to solve it. i know im missing something simple. so can anyone help me solve this please. thank you

**e^(i*pi)** · October 1st 2011, 10:08 AM

Originally Posted by salma2011

I can't remember how to solve it. i know im missing something simple. so can anyone help me solve this please. thank you

I suggest finding the lowest common denominators of 3 and 5 and change your exponents into that form:

$f'(x) = \dfrac{4}{3}x^{-5/15} + \dfrac{4}{5}x^{-3/15} = 0$

You can then factor out $4x^{-1/15}$ as a common factor

$4x^{-3/15}\left(\dfrac{x^2}{3} + \dfrac{1}{5}\right) = 0$

Can you solve now?

To test whether you have a maximum or minimum check the sign of $f''(x)$ .

If $f''(x) < 0$ you have a maximum
If $f''(x) > 0$ you have a minimum
If $f''(x) = 0$ you have a point of inflection

**Siron** · October 1st 2011, 10:13 AM

There're probably different ways to solve the equation, but if you have to solve:
$0=\frac{4}{3}x^{\frac{-1}{3}}+\frac{4}{5}x^{\frac{-1}{5}}$
$\Leftrightarrow 4x^{\frac{-1}{3}}\left(\frac{1}{3}+\frac{1}{5}x^{\frac{2}{15} }\right)=0$
$\Leftrightarrox x^{\frac{-1}{3}}=0 \ \mbox{or} \ x^{\frac{2}{15}}=\frac{-5}{3}$

This means there're no solutions, because there's no $x$ value wherefore:
$x^{\frac{-1}{3}}=0$ or $x^{\frac{2}{15}}=\frac{-5}{3}$ .

So what's your conclusion?

**salma2011** · October 1st 2011, 10:59 AM

i solved for x and got x = + or - square root of 3/5

(sorry don't know how to put mathematical functions onto the computer)

and then from the derivative i work out whether it's a maximum or a minimum.

thank you for all your help

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