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Math Help - Maxima/Mininima

  1. #1
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    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 ^_^
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  2. #2
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    First, differentiate f(x,y), then find where f'(x,y) equals zero. Those are your potential minimums and maximums.

    For example:

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

    y' = x^2 + x - 12

    x^2 + x - 12 = 0

    (x - 3)(x + 4) = 0

    y' = 0 at x = 3 and x = -4

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

    y'' = 2x + 1

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

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


    Scott
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ScottO View Post
    First, differentiate f(x,y), then find where f'(x,y) equals zero. Those are your potential minimums and maximums.

    For example:

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

    y' = x^2 + x - 12

    x^2 + x - 12 = 0

    (x - 3)(x + 4) = 0

    y' = 0 at x = 3 and x = -4

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

    y'' = 2x + 1

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

    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.

    Quote Originally Posted by MarianaA View Post
    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 f(x,y) is continuous on some disk centered at the point (a,b).

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

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

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

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

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

    case 3: If D<0, then f(a,b) is a saddle point
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  4. #4
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    Quote Originally Posted by Jhevon View Post
    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
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MarianaA View Post
    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?
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  6. #6
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    Quote Originally Posted by Jhevon View Post
    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 didnít believe me, but it is true.
    Cost me around $120 dollars, so probably they already sell it,
    you never know.
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  7. #7
    Forum Admin topsquark's Avatar
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    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
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MarianaA View Post
    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

    Quote Originally Posted by topsquark View Post
    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)
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