second order differentiations stationary points problems on maxima and minima.

**mastermin346** · June 5th 2010, 11:53 PM

Two positive quantities, $p$ and $q$ ,very in such a way that ${p^3}{q}=9$ .Another quantity $z$ ,is defined by $z=16p+3q$ .Find the values of $p$ and $q$ that makes $z$ a minimum.

help.

**SpringFan25** · June 6th 2010, 04:14 AM

you can use Lagrange multipliers - Wikipedia, the free encyclopedia to do this.

$\min_{p,q} L = 16p + 3q +\lambda \left(p^3q - 9 \right)$
To solve the optimisation problem, differenciate L with respect to p,q and $\lambda$ and set all partial derivatives to zero.

**skeeter** · June 6th 2010, 04:18 AM

Originally Posted by mastermin346

Two positive quantities, $p$ and $q$ ,very in such a way that ${p^3}{q}=9$ .Another quantity $z$ ,is defined by $z=16p+3q$ .Find the values of $p$ and $q$ that makes $z$ a minimum.

$q = \frac{9}{p^3}$

$z = 16p + \frac{27}{p^3}$

you've posted this in the precalculus section ... therefore I assume you'll be using technology (rather than calculus) to determine the value of p that minimizes z.

**SpringFan25** · June 7th 2010, 02:43 PM

lol i cant believe i forgot that...skeeters way is much better

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