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Math Help - Finding 1st and 2nd partial derivitives

  1. #1
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    Finding 1st and 2nd partial derivitives

    The starting function I have is f=(1/6)x^3 + 2xy^2 + xy
    I need to find the 1st and 2nd partial derivatives, and hence find the location and nture of the four stationary points. How do I go about doing that?
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  2. #2
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    Hi

    To find the partial derivative with respect to x of f(x) = \frac{1}{6}\:x^3 + 2xy^2 + xy you must consider y as constant

    \frac{\partial f}{\partial x} = \frac{1}{6}\:3\:x^2 + 2y^2 + y = \frac{1}{2}\:x^2 + 2y^2 + y

    Same method to calculate \frac{\partial f}{\partial y} : consider x as constant
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  3. #3
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    f=\frac{1}{6}x^3+2xy^2+xy

    Partial differentiation basically involves choosing a variable to differentiate with respect to, and treating all others as constants. So:

    \frac{\partial f}{\partial x}=f_x=\frac{1}{2}x^2+2y^2+y

    \frac{\partial f}{\partial y}=f_y=4x+x=5x

    I leave you to find

    \frac{\partial ^2f}{\partial x^2}=f_{xx}
    \frac{\partial ^2f}{\partial x\partial y}=f_{xy}=f_{yx}=\frac{\partial ^2f}{\partial y\partial x}
    \frac{\partial ^2f}{\partial y^2}=f_{yy}
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  4. #4
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    Quote Originally Posted by Thomas154321 View Post
    \frac{\partial f}{\partial y}=f_y=4x+x=5x
    Sorry but there is a little mistake

    \frac{\partial f}{\partial y}=f_y=4xy+x
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  5. #5
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    Still need to find stat. points

    Ok I see where you're coming from, as I basically did the same thing before I made this post. I just needed to see if my working as correct. The only problem now for me was to find the stationary points and their nature, i.e. max or min. This is where I am stuck, the first part was pretty easy.

    Quote Originally Posted by Thomas154321 View Post
    f=\frac{1}{6}x^3+2xy^2+xy

    Partial differentiation basically involves choosing a variable to differentiate with respect to, and treating all others as constants. So:

    \frac{\partial f}{\partial x}=f_x=\frac{1}{2}x^2+2y^2+y

    \frac{\partial f}{\partial y}=f_y=4x+x=5x

    I leave you to find

    \frac{\partial ^2f}{\partial x^2}=f_{xx}
    \frac{\partial ^2f}{\partial x\partial y}=f_{xy}=f_{yx}=\frac{\partial ^2f}{\partial y\partial x}
    \frac{\partial ^2f}{\partial y^2}=f_{yy}
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  6. #6
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    Hello, LooNiE!

    The starting function is: . f(x,y)\:=\:\tfrac{1}{6}x^3 + 2xy^2 + xy
    I need to find the 1st and 2nd partial derivatives,
    and hence find the location and nature of the four stationary points.
    How do I go about doing that? . .
    And no one has told you how?
    To find the stationary points, set the first partials equal to zero and solve.

    . . . \begin{array}{ccccccc}f_x &=&\frac{1}{2}x^2 + 2y^2 + y &=&0 & {\color{red}[1]} \\ \\[-3mm]<br />
f_y &=&4xy + x &=&0 & {\color{red}[2]}\end{array}


    From [2], we have: . x(4y+1) \:=\:0 \quad\Rightarrow\quad x = 0,\:y = \text{-}\tfrac{1}{4}

    Substitute x=0 into [1]: . 2y^2 + y \:=\:0 \quad\Rightarrow\quad y \:=\:0,\:\text{-}\tfrac{1}{2}
    . . We have two of the points: . {\color{blue}(0,0),\:\left(0,\text{-}\tfrac{1}{2}\right)}

    Substitute y = \text{-}\tfrac{1}{4} into [1]: . \tfrac{1}{2}x^2 + \tfrac{1}{8} -\tfrac{1}{4} \:=\:0 \quad\Rightarrow\quad x = \pm\tfrac{1}{2}
    . . And we have two more points: . {\color{blue}\left(\tfrac{1}{2},\:\text{-}\tfrac{1}{4}\right),\:\left(\text{-}\tfrac{1}{2},\:\text{-}\tfrac{1}{4}\right)}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    For the Second Partial Test, form the expression: . D \;=\;\left(f_{xx}\right)\left(f_{yy}\right) - \left(f_{xy}\right)^2

    We have: . f_{xx}\:=\:x,\;\;f_{yy}\:=\:4x,\;\;f_{xy} \:=\:4y+1

    . . Then: . D \;=\;4x^2 - (4y+1)^2


    At {\color{blue}(0,0)}\!:\; D\:=\:4(0) - (0+1)^2 \:=\:-1 . . . negative: saddle point

    At {\color{blue}\left(0,\text{-}\tfrac{1}{2}\right)}\!:\;D \:=\:4(0) - (\text{-}2+1)^2 \:=\:-1 . . . negative: saddle point

    At {\color{blue}\left(\tfrac{1}{2},\text{-}\tfrac{1}{4}\right)}\!:\;D\:=\:4(\tfrac{1}{4}) - (-1+1)^2 \:=\:+1
    . . . f_{xx}  \:=\:+\tfrac{1}{2} . . . positive, concave up, \cup . . . minimum

    At {\color{blue}\left(\text{-}\tfrac{1}{2},\text{-}\tfrac{1}{4}\right)}\!:\;D \:=\:4(\tfrac{1}{4}) - (-1+1)^2 \:=\:+1
    . . . f_{xx} \:=\:\text{-}\tfrac{1}{2} . . . negative, concave down, \cap . . . maximum

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