# second order differentiations stationary points problems on maxima and minima.

• Jun 5th 2010, 11:53 PM
mastermin346
second order differentiations stationary points problems on maxima and minima.
Two positive quantities,$\displaystyle p$ and $\displaystyle q$,very in such a way that $\displaystyle {p^3}{q}=9$.Another quantity $\displaystyle z$,is defined by $\displaystyle z=16p+3q$.Find the values of $\displaystyle p$ and $\displaystyle q$ that makes $\displaystyle z$ a minimum.

help.
• Jun 6th 2010, 04:14 AM
SpringFan25
you can use Lagrange multipliers - Wikipedia, the free encyclopedia to do this.

$\displaystyle \min_{p,q} L = 16p + 3q +\lambda \left(p^3q - 9 \right)$
To solve the optimisation problem, differenciate L with respect to p,q and $\displaystyle \lambda$ and set all partial derivatives to zero.
• Jun 6th 2010, 04:18 AM
skeeter
Quote:

Originally Posted by mastermin346
Two positive quantities,$\displaystyle p$ and $\displaystyle q$,very in such a way that $\displaystyle {p^3}{q}=9$.Another quantity $\displaystyle z$,is defined by $\displaystyle z=16p+3q$.Find the values of $\displaystyle p$ and $\displaystyle q$ that makes $\displaystyle z$ a minimum.

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

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

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.
• Jun 7th 2010, 02:43 PM
SpringFan25
lol i cant believe i forgot that...skeeters way is much better :D