Determining the Roots of an Equation with Two Real Solutions

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SUMMARY

The discussion centers on determining the roots of the polynomial equation x^4 + 3x^3 - 2 = 0, specifically proving that it has exactly two real roots. The user also seeks assistance in differentiating the packing fraction formula f(x) = K(1+c^2x^3)/(1+x)^3 using Maple software. The conversation highlights the application of the Mean Value Theorem (MVT) in root-finding and the need for differentiation techniques in crystallography-related calculations.

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  • Understanding of polynomial equations and real roots
  • Familiarity with the Mean Value Theorem (MVT)
  • Basic knowledge of differentiation techniques
  • Experience with Maple software for mathematical computations
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  • Learn how to use Maple for symbolic differentiation
  • Study the application of the Mean Value Theorem in proving the existence of real roots
  • Explore the concept of packing fractions in crystallography
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Mathematicians, crystallographers, students studying calculus, and anyone interested in polynomial root-finding techniques and differentiation in mathematical software.

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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