# Minima and Partial Derivatives

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 $$\frac{\delta^2 A}{\delta x \delta t}$$?

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.