1. Maxima/Mininima

Hi, can some one help with this please.

How I can fine all of the potential maxima and minima for f(x,y).

Thanks ^_^

2. First, differentiate f(x,y), then find where f'(x,y) equals zero. Those are your potential minimums and maximums.

For example:

$\displaystyle y = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x + 6$

$\displaystyle y' = x^2 + x - 12$

$\displaystyle x^2 + x - 12 = 0$

$\displaystyle (x - 3)(x + 4) = 0$

$\displaystyle y' = 0$ at $\displaystyle x = 3$ and $\displaystyle x = -4$

To determine which they are, find y'' and plug in the zero values for x...

$\displaystyle y'' = 2x + 1$

$\displaystyle y''(3) = 2(3) + 1 = 7 > 0$ so minimum

$\displaystyle y''(-4) = 2(-4) + 1 = -7 < 0$ so maximum

Scott

3. Originally Posted by ScottO
First, differentiate f(x,y), then find where f'(x,y) equals zero. Those are your potential minimums and maximums.

For example:

$\displaystyle y = \frac{1}{3}x^3 + \frac{1}{2}x^2 - 12x + 6$

$\displaystyle y' = x^2 + x - 12$

$\displaystyle x^2 + x - 12 = 0$

$\displaystyle (x - 3)(x + 4) = 0$

$\displaystyle y' = 0$ at $\displaystyle x = 3$ and $\displaystyle x = -4$

To determine which they are, find y'' and plug in the zero values for x...

$\displaystyle y'' = 2x + 1$

$\displaystyle y''(3) = 2(3) + 1 = 7 > 0$ so minimum

$\displaystyle y''(-4) = 2(-4) + 1 = -7 < 0$ so maximum

Scott
I believe the poster was talking about functions of two variables. it is a slightly different.

Originally Posted by MarianaA
Hi, can some one help with this please.

How I can fine all of the potential maxima and minima for f(x,y).

Thanks ^_^
this is the kind of question for which you should consult your text for the answer.

We suppose the second partial derivatives of the function $\displaystyle f(x,y)$ is continuous on some disk centered at the point $\displaystyle (a,b)$.

If $\displaystyle f_x(a,b) = 0$ and $\displaystyle f_y(a,b) = 0$ then we call $\displaystyle (a,b)$ a critical point of $\displaystyle f$.

Define $\displaystyle D(a,b) = f_{xx}(a,b)f_{yy}(a,b) - [ f_{xy}(a,b)]^2$

we can classify the critical point $\displaystyle (a,b)$ as follows:

case 1: If $\displaystyle D>0$ and $\displaystyle f_{xx}(a,b)>0$, then $\displaystyle f(a,b)$ is a local minimum

case 2: If $\displaystyle D>0$ and $\displaystyle f_{xx}(a,b)<0$, then $\displaystyle f(a,b)$ is a local maximum

case 3: If $\displaystyle D<0$, then $\displaystyle f(a,b)$ is a saddle point

4. Originally Posted by Jhevon
this is the kind of question for which you should consult your text for the answer.
I know but some one steal my book, thats why I am asking.
Thanks both of you

5. Originally Posted by MarianaA
I know but some one steal my book, thats why I am asking.
Thanks both of you
what monster would steal a calculus book? what is the world coming to?!

bless your heart Mariana, you have my condolences

i shudder to think what i'd do if someone stole my calculus text

did you understand the notation i used?

6. Originally Posted by Jhevon
what monster would steal a calculus book? what is the world coming to?!

bless your heart Mariana, you have my condolences

i shudder to think what i'd do if someone stole my calculus text

did you understand the notation i used?
Looks like you didn’t believe me, but it is true.
Cost me around $120 dollars, so probably they already sell it, you never know. 7. My Calc I professor and I wound up switching books during a class one day. (Unfortunately, no it wasn't a "teacher's" text with the answers.) I didn't find out about it until after finals, when noticed some scribblings in the part of the text I had read earlier that semester. Since I had bought the book new and didn't make the scribbles (that weren't there when I read the material) I realized what had happened. I paid for a new textbook and wound up with a used and written in book! (If that story doesn't move you, then you obviously don't love and worship your textbooks like I do.) -Dan 8. Originally Posted by MarianaA Looks like you didn’t believe me, but it is true. Cost me around$120 dollars, so probably they already sell it,
you never know.
i believe you

Originally Posted by topsquark
My Calc I professor and I wound up switching books during a class one day. (Unfortunately, no it wasn't a "teacher's" text with the answers.) I didn't find out about it until after finals, when noticed some scribblings in the part of the text I had read earlier that semester. Since I had bought the book new and didn't make the scribbles (that weren't there when I read the material) I realized what had happened.

I paid for a new textbook and wound up with a used and written in book!

(If that story doesn't move you, then you obviously don't love and worship your textbooks like I do.)

-Dan
i am moved. you did get your book back once you realized though, right? (on second thought, don't answer that. all these off topic posts are bound to make TPH angry)