I need to say which plane goes through the point (3,5,6), intersects with the positive side of all axis and creates the pyramid with the minimal volume with those axis.
I'm not sure how to begin this.
I need to say which plane goes through the point (3,5,6), intersects with the positive side of all axis and creates the pyramid with the minimal volume with those axis.
I'm not sure how to begin this.
When the axis intercepts of your plane are $a$, $b$, $c>0$ then the volume of said pyramid is given by $V={1\over 6} abc$. The equation of this plane can be written in the form ${x\over a}+{y\over b}+{z\over c}=1\ ,$ and as it has to pass through the point $(3,5,6)$ we have the extra condition ${3\over a}+{5\over b}+{6\over c}=1\ .\qquad(1)$ Therefore we have to minimize $abc$ given the side condition $(1)$. This can be done with the help of Lagrange multipliers or with the help of a convexity argument.
adding to what Christian Blatter said, we can observe that harmonic mean is less than geometric mean always, and the G.M == H.M only when all the numbers are equal. which would mean 3/a=1/3, 5/b=1/3 and 6/c=1/3