- #1

Hollysmoke

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A fundamental problem in crystallography is the determination of the packing fraction of a crystal lattice, which is the fraction of space occupied by the atoms in the lattice, assuming that the atoms are hard spheres. When the lattice contains exactly two different kinds of atoms, it can be shown that the packing fraction is given by the formula:

f(x) = K(1+c^2x^3)/(1+x)^3

where x=r/R is the ratio of the radii, r and R of the two kinds of =atoms in the lattice, and c and K are positive constants.

How can I input this into maple to differentiate it?

And also:

x^4+3x^3-2=0

I'm supposed to prove that there are exactly 2 real roots. I tried using MVT but I'm not getting the answer =(