# families of curves --- help!

• Nov 20th 2007, 01:17 PM
Paige05
families of curves --- help!
heres the question:

find a formula for a function of the form y = (ax^b)(lnx) where a and b are nonzero constants, which has a local maximum at the point (e^2,6e^-1)

i tried to create 2 equations and set and solve for the variables, but its giving me a rediculous answer so i dont think its right.

anyone know what to do?
• Nov 20th 2007, 07:14 PM
TKHunny

You are right, though. The answers are not all that complicated.
• Nov 21st 2007, 08:16 AM
topsquark
Quote:

Originally Posted by Paige05
heres the question:

find a formula for a function of the form y = (ax^b)(lnx) where a and b are nonzero constants, which has a local maximum at the point (e^2,6e^-1)

i tried to create 2 equations and set and solve for the variables, but its giving me a rediculous answer so i dont think its right.

anyone know what to do?

$y^{\prime}(x) = abx^{b - 1}ln(x) + ax^{b - 1}$

You wish that
$y^{\prime}(e^2) = ab(e^2)^{b - 1}ln(e^2) + a(e^2)^{b - 1} = 0$

And also
$6e^{-1} = a(e^2)^b \cdot ln(e^2)$ <-- From your original equation.

These two equations simplify to
$2abe^{2(b - 1)} + ae^{2(b - 1)} = 0$
and
$\frac{6}{e} = 2ae^{2b}$

Can you solve these for a and b? (Note that you will also need to show that this is a local maximum, not a local minimum.)

-Dan
• Nov 21st 2007, 11:29 AM
Paige05
those are exactly the equations that i came up with, but when i solved for a and put that back into the second equation, i find it so hard to simplify and solve for b. im stuck.
• Nov 21st 2007, 11:35 AM
Paige05
nevermind! i got it! :)

thank you :)