Answer
Sum of the intercepts is a constant.
Work Step by Step
Formula to calculate tangent plane equation is:
$(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$
At point$(x_1,y_1,z_1)$
$\dfrac{(x-x_1)}{(2\sqrt{x_1})}+\dfrac{(y-y_1)}{(2\sqrt{y_1})}+\dfrac{(z-z_1)}{(2\sqrt{z_1})}=0$
$\dfrac{(x-x_1)}{(\sqrt{x_1})}+\dfrac{(y-y_1)}{(\sqrt{y_1})}+\dfrac{(z-z_1)}{(\sqrt{z_1})}=0$
Since, $(x_1,y_1,z_1)$ is the point which satisfies $\sqrt x+\sqrt y+\sqrt z=\sqrt c$
Hence, the sum of the intercepts is a constant.
