Let fxy0 be continuous and fx0fy0 for all xy∈R2 Let I1=∫20∫10fxydydx I2=∫32∫10fxydydx I3=∫20∫31fxydydx I4=∫32∫31fxydydx Then which of the following is always true

Let's draw the region of integration:

We can see that $I_1+I_2>I_3+I_4$, since the rectangles corresponding to $I_1$ and $I_2$ are larger than those corresponding to $I_3$ and $I_4$, respectively.

Now, using the fact that $f$ is positive and decreasing in each variable, we can see that $I_1>I_2$ and $I_3>I_4$. Therefore, we have \begin{align*} I_1+I_2&>I_3+I_4 \ \Rightarrow I_1&>I_3+I_4-I_2. \end{align*}Thus, $\textbf{(A)}$ is always true.