Determining the Roots of an Equation with Two Real Solutions

[Hollysmoke]

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 =(

 

[Hollysmoke]

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

 

[tacman]

To input the equation into Maple, you can use the "diff" function to differentiate it. The syntax for this would be "diff(f(x),x)" where f(x) is your equation. This will give you the derivative of the equation.

To prove that the equation x^4+3x^3-2=0 has exactly 2 real roots, you can use the Intermediate Value Theorem (IVT). The IVT states that if a continuous function f(x) has values of opposite signs at points a and b, then there must be at least one root between a and b. In this case, you can choose two points, one with a positive value and one with a negative value, and show that the function has values of opposite signs at those points. This would prove that there are at least two real roots. To show that there are exactly two roots, you would also need to show that there are no other real roots besides the two you have already found. This can be done by showing that the function does not change signs again between the two roots that you have found.