Answer
The volume of the pyramid cut off from the first octant by any tangent plane to the surface $xyz=1$ at points in the first octant is the same.
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$(a,b,c)$
$(x-a)bc)+(y-b)ac+(z-c)ab=0$
$3abc=bcx+acy+abz$
We need to find the x-,y-,x- intercepts.
x-intercept: 3a; y-intercept: 3b and z-intercept: 3c
Volume of the pyramid is given as:$V=\dfrac{1}{3}(\dfrac{9ab}{2})(3c)=\dfrac{9abc}{2}$
Since, $xyz=1 \implies abc=1$
Thus, $V=\dfrac{9}{2}$
This has been proved. The volume of the pyramid cut off from the first octant by any tangent plane to the surface $xyz=1$ at points in the first octant is the same.
