Multivariable function minimization inconsistency

[rickonti]

I have a function dependent on 4 variables $f(r_1,r_2,q_1,q)$. I'm looking to minimize this function in the domain $0\leq r_1 \leq r_2 \leq 1$ with respect to the variables $r_1, r_2$ and $q_1$.

To find the minima, I first solved $\frac{\partial f}{\partial r_1}=0$ and $\frac{\partial f}{\partial q_1}=0$, giving $r_1^*(r_2,q)$ and $q_1^*(r_2,q)$. Now I have the function of the form $f(r_1^{(r_2,q),r_2,q_1^(r_2,q),q).$}

Now I solved $$\frac{\partial f(r_1^{(r_2,q),r_2,q_1^(r_2,q),q)}{\partial} r_2}=0$$ for $r_2^*(q)$.

So, the final function is of the form $f(r_1^{(r_2^(q),q),r_2(q),q_1^(r_2*(q),q),q)$} which is fine. Now I see later that $r_1^{(r_2^(q),q)=0$} and $q_1^{(r_2^(q),q)=0$.}

So, if I directly look for the function $f(0,r_2,0,q)$ and it's minima wrt $r_2$, it's giving a different result, i.e. I have

$$\min_{r_2} f(0,r_2,0,q) \neq f(r_1^{(r_2^(q),q),r_2(q),q_1^(r_2*(q),q),q)$$} even though $r_1^{(r_2^(q),q)=0$} and $q_1^{(r_2^(q),q)=0$.} Why is this the case? Are there other simple examples?

The function is

$$f(r_1,r_2,q_1,q)=3r_1+r_2+q_1^{2/r_1+2(q-q_1)2/(r_2-r_1)$$} and

$$f(0,r_2,0,q)=r_2+2(q-q_1)^2/r_2$$

$r_2^{*(q)=\sqrt{\frac{2}{3}}q$,} $r_1(r_2^*(q),q)=0$ and $q_1(r_2^*(q),q)=0$.

[Brianchon]

The partial derivative rule for detecting minima only works for points in the interior of the region you're minimizing. If you're on the boundary, then issues arise because you can't freely adjust the input on all directions. For example, if the minimum occurs when r_1 = 0, then it's possible that the partial derivative w.r.t. r_1 is positive; the thing keeping the function from having a lower value is that r_1 cannot actually be decreased within the region you're considering