Hi, can some one help with this please.
How I can fine all of the potential maxima and minima for f(x,y).
Thanks ^_^
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.
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
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