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Thread: Find the stationary points.

  1. #1
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    Find the stationary points.

    Find the stationary points of this function f(x)=x^3 - 6xy + 3y^2

    Can someone please help me with calculus problem? (detailed please)
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  2. #2
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    Your function is $\displaystyle \displaystyle f(x,y) = x^3 - 6xy + 3y^2$.

    The gradient vector is $\displaystyle \displaystyle \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$ and the critical points occur when $\displaystyle \displaystyle \nabla f = \mathbf{0}$.


    $\displaystyle \displaystyle \nabla f = \left(3x^2 - 6y, -6x + 6y\right) = \mathbf{0}$.


    So $\displaystyle \displaystyle 3x^2 - 6y = 0$ and $\displaystyle \displaystyle -6x + 6y = 0$.

    From the second equation, $\displaystyle \displaystyle x = y $.

    Substituting into the first gives $\displaystyle \displaystyle 3x^2 - 6x = 0$

    $\displaystyle \displaystyle 3x(x - 2) = 0$

    $\displaystyle \displaystyle x = 0, x = 2$.


    Therefore, the critical points are $\displaystyle \displaystyle (x, y) = (0, 0)$ and $\displaystyle \displaystyle (x, y) = (2, 2)$.
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  3. #3
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    Thank you so much.
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