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How to find the Critical Points of a Multivariable Function
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How to find and classify the critical points of multivariable functions.
Begin by finding the partial derivatives of the multivariable function with respect to x and y. Next find the second order partial derivatives fxx, fyy and fxy. From here, the critical points can be found by setting fx and fy equal to 0 and solving the subsequent simultaneous equation for x and y. Once you have found the critical points, the next step is to find a value for the discriminant and use the second partial derivative test to establish if the critical point is a local minimum, local maximum, saddle point or if the test is inconclusive.
Music by Adrian von Ziegler
Begin by finding the partial derivatives of the multivariable function with respect to x and y. Next find the second order partial derivatives fxx, fyy and fxy. From here, the critical points can be found by setting fx and fy equal to 0 and solving the subsequent simultaneous equation for x and y. Once you have found the critical points, the next step is to find a value for the discriminant and use the second partial derivative test to establish if the critical point is a local minimum, local maximum, saddle point or if the test is inconclusive.
Music by Adrian von Ziegler
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