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I am dealing with an equation that has two variables x and t. I have the following equation for the first order partial derivative with respect to x:

$\frac{\delta A}{\delta x} = \frac{2t - 3t^2}{2xt - 3xt^2}$.

Setting this equal to 0, I got a value for t. I can see it has 2 roots. Now, I wanted to check if any of these t achieves maxima or minima. When I compute the second derivative, I get the following:

$\frac{\delta^2 A}{\delta^2 x} = (-1) \frac{(2t - 3t^2)^2}{(2xt - 3xt^2)^2}$

This term is always negative. So, does it mean that this t value never achieves the minima? Or should I do $\frac{\delta^2 A}{\delta x \delta t}$?

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Since $A$ depends on both $x$ and $t$, the extrema are found first by setting $\frac{\partial A}{\partial x} = 0$ AND $\frac{\partial A}{\partial t} = 0$. Both of these equations must be satisfied for a point $(a,b)$ to be an extrema.

To check if this point you found is a max or min, you want to compute the Hessian $$H(a,b) = A_{xx}(a,b)A_{tt}(a,b) - 2A_{xt}(a,b)$$ at $(a,b)$. If $H(a,b)>0$, then $(a,b)$ is a relative maximum if $A_{xx}(a,b)<0$, or a minimum if $A_{xx}(a,b) >0$. If $H(a,b) < 0$, $(a,b)$ is a saddle point.

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