(a) well, you know critical numbers are where f'(x) is zero or undefined.
we can factor out the K and ignore it, since it is just a scale factor. Using
f(x) = (1+c^2x^3)/(1+x)^3
f' =
(3c^2x^2)(1+x)^3 - (1+c^2x^3)(3)(1+x)^2
----------------------------------------------
(1+x)^6
or, factoring out the (a+x)^2,
(1+c^2x^3)/(1+x)^3
f' =
(3c^2x^2)(1+x) - 3(1+c^2x^3)
---------------------------------
(1+x)^4
f'(x) is undefined at x = -1
f'(x)=0 when the numerator is zero. That is, when
(3c^2x^2)(1+x) - 3(1+c^2x^3) = 0
3(cx-1)(cx+1) = 0
x = ±1/c
Maybe this will get you started. Gotta hit the rack now, but I'll check back in in a few hours.
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^2*x^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 K and c are positive constants.
(a) The function f(x) has exactly one critical number. Find it and use the second
derivative test to classify it as a relative maximum or a relative minimum.
(b) The numbers c and K and the domain of f(x) depend on the cell structure in the lattice. For ordinary rock salt: c = 1, K = (2π/3), and the domain is the interval
(sqrt(2) − 1) <= x ≤<=1. Find the largest and smallest values of f(x).
(c) Repeat part (b) for β-cristobalite, for which c = sqrt(2), K = sqrt(3π/16), and the domain is 0 <= x <= 1.
(d) What can be said about the packing fraction f(x) if r is much larger than R?
Answer this question by computing lim f(x).
x→∞
Hey! Sorry that I am confused on this question. I don't know where to start. Can someone help me how to work through the problem? I am just lost on what's going on here. I think I am that type of person who likes to work backwards to see how to do this problem.
Thank you in advance! :)
2 answers
I'm working on the same problem, any chance you could help me through the second derivative test section?
2 answers