separable utility function and cross price effect Suppose that consumer's utility function for three goods is separable, that is,

$U(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ ...(i)

where $f_i$ is increasing and strictly concave, i=1,2,3. Show that



$ df_i/dP_1 $ for i = 2,3 can be either positive or negative, but they must be of the same sign where $f_i$ is the demand function for good i and $P_1$ is price of good 1.



i reasoned using the equimarginal principle where at optimum we must have 

$ MU_1/P1 = MU_2/P2 = MU_3/P3 $.

suppose good 1 is normal, if $P_1$ rises demand for good 1 falls, that is $df_1/dP_1$ &lt; 0. Due to law of diminishing marginal utility $MU_1$ rises, however to maintain equilibrium $ MU_2,  MU_3$ must also rise. So, $df_2/dp_1 &lt; 0$ and $ df_3/dP_1 &lt; 0$. But i couldn't do it more mathematically, taking the derivative of (i) w.r.t $P_1$ isn't of much use, please help.
The following argument assumes that we are dealing only with interior solutions. Let $(x_1,x_2,x_3)$ be an optimal demand bundle at prices $(p_1,p_2,p_3)$ and income $m$ and assume that $(x_1,x_2,x_3)$ is an optimal demand bundle at prices $(p_1',p_2,p_3)$ and income $m$. We want to show that one of the following three cases holds: 


$x_2=x_2'$ and $x_3=x_3'$,
$x_2&lt;x_2'$ and $x_3&lt;x_3'$, or
$x_2&gt;x_2'$ and $x_3&gt;x_3'$.


What is not allowed to happen is that $x_2\leq x_2'$ and $x_3&gt;x_3'$ or $x_2&gt;x_2'$ and $x_3\leq x_3'$. Assume one of these cases happens, without loss of generality, we can assume it is the first.

By the strict concavity of $f_2$ and $f_3$, we have
$$MU_2(x_1,x_2,x_3)=\frac{\partial f_2(x_2)}{\partial x_2}\geq \frac{\partial f_2(x_2')}{\partial x_2}=MU_2(x_1',x_2',x_3')$$
and
$$MU_3(x_1,x_2,x_3)=\frac{\partial f_3(x_3)}{\partial x_3}&lt; \frac{\partial f_3(x_3')}{\partial x_3}=MU_3(x_1',x_2',x_3').$$ 
Therefore,
$$\frac{MU_2(x_1',x_2',x_3')}{p_2}\leq\frac{MU_2(x_1,x_2,x_3)}{p_2}=\frac{MU_3(x_1,x_2,x_3)}{p_3}&lt;\frac{MU_3(x_1',x_2',x_3')}{p_3},$$
in contradiction to the equimarginal principle.