# Finding maximum

• Feb 22nd 2008, 07:56 AM
XIII13Thirteen
Finding maximum
$\displaystyle y=3x-\frac{x^2}{2}$

Find y max

Here is how I'm approaching this ...

$\displaystyle 6x-x^2$

$\displaystyle y'=6-2x$

With 3 being a critical point

$\displaystyle y''=-2$

So the critical point is a maximum. It's a parabola opening downward from 3?
• Feb 22nd 2008, 08:52 AM
wingless
We have $\displaystyle y = 3x - \frac{x^2}{2}$

But
$\displaystyle 6x-x^2$ ??
$\displaystyle y' = 6-2x$ ??
These are wrong.

But I see what you were trying to do.

It should be,

$\displaystyle y = 3x - \frac{x^2}{2}$

$\displaystyle 2y = 6x - x^2$

$\displaystyle 2y' = 6 - 2x$

$\displaystyle y' = 3 - x$

I know, the root doesn't change by multiplication or division, but $\displaystyle y' = 6-2x$ is not correct.
• Feb 22nd 2008, 09:24 AM
XIII13Thirteen
Thank you for clarifying. I'm prone to making mistakes and that is why I like to double check. Looking at the equation is still seems 3 is our critical point, but the differences in our y' will become ...

$\displaystyle y''=-1$

Which should still give us 3 as our maximum with a downward opening parabola. I appreciate the help as this mistake could have cost me in other equations
• Feb 22nd 2008, 09:45 AM
galactus
Are you just trying to find the max of $\displaystyle 3x-\frac{x^{2}}{2}$?.

Did you just differentiate?.

$\displaystyle y'=3-x$

3-x=0, x=3

That gives y=9/2 as the max value for y.

If I am misunderstanding, I apologize.
• Feb 22nd 2008, 09:54 AM
wingless
Quote:

Originally Posted by XIII13Thirteen
Which should still give us 3 as our maximum with a downward opening parabola. I appreciate the help as this mistake could have cost me in other equations

Do you know that the maximum is at $\displaystyle x=3$ and the maximum value of the function is $\displaystyle f(3) = 9/2$ as galactus said? If you know it, no problem (Wink)
• Feb 22nd 2008, 09:59 AM
galactus
You can also find this without calc by using $\displaystyle x=\frac{-b}{2a}$.

Which gives the x-coordinate for the vertex of a parabola.

In your case, a=-1/2 and b=3

$\displaystyle \frac{-3}{2(\frac{-1}{2})}=3$
• Feb 22nd 2008, 10:09 AM
XIII13Thirteen
The equation wanted "y max," not "x max" so technically it is $\displaystyle \frac{9}{2}$ is correct. I'm doing these equations with a multiple choice solution set and he has
I originally picked b because I was thinking of in terms of x, but in the context of "find y max" $\displaystyle \frac{9}{2}$. Evil questions. Thanks for catching that