How to use clairaut's theorem with 3rd order partial derivatives

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To apply Clairaut's Theorem for third-order partial derivatives, it is essential to establish that the mixed partial derivatives are continuous. The theorem states that if the second-order mixed derivatives are continuous, then the order of differentiation does not affect the result, leading to the equality fxxy = fyxy = fyyz. An attempt to differentiate the second partial derivatives was made, but it was unsuccessful, prompting a counterexample using the function f = (x^3)(y^2)(z). The specific values (5, 7, 11) were suggested for testing the function's behavior. Understanding the continuity of these derivatives is crucial for applying Clairaut's Theorem effectively.
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Homework Statement



Use Clairaut's Theorem to show that is the third order partial derivatives are continuous, then fxxy=fyxy=fyyz

Clairaut's Theorem being: fxy(a,b)=fyx(a.b)

Homework Equations



fxyy=d/dy(d2f/dydx)=d^3f/dy^2dx

The Attempt at a Solution



Tried to differentiate the second partial derivatives but it didn't work out.
 
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counter example

Try when f = (x^3)(y^2)(z). After differentiating, try (5,7,11).
 

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