Determining the Roots of an Equation with Two Real Solutions

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The discussion revolves around two main questions related to mathematical problems in crystallography and polynomial equations. The first question involves differentiating a packing fraction formula in crystallography, specifically f(x) = K(1+c^2x^3)/(1+x)^3, where x represents the ratio of the radii of two types of atoms in a crystal lattice. The user seeks guidance on how to input this formula into Maple for differentiation. The second question pertains to a polynomial equation, x^4 + 3x^3 - 2 = 0, where the user aims to prove that there are exactly two real roots. Although the user has made progress on this problem using the Mean Value Theorem (MVT), they are still seeking assistance with the differentiation aspect of the first question.
Hollysmoke
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I just have two questions:

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 =(
 
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I figured out the 2nd one ^^ Just need help with the differentiation one.
 

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