Thus, rli, r2i, r3i and r4i are the local minima or
local maxima of r expressed by Eq. (3.9). In order to find
the global minimum and maximum values, we take the second
derivative of r with respect to 9bi of Eq. (3.10), which
yields
d2r dr
r + (- )2 = -bi[biS2abi(2C2 bi 1) +
debi2 d8bi
aiCebi] (3.16)
At the position of local minima or maxima, the first
derivatives equal zero, and then
d2r
r
debi2 ebi
d2r
bi2
d8bi2 ebi
= -bi(biS2abi + ai) < 0
= -bi(bis2abi ai) < 0
(3.17)
(3.18)
At the position of r3 or r4,
d2r
r ----- -ai
drbi2 Gbi = arc cos ( --
biS2 bi
ai2 bi2S4abi
S2,bi
> 0 (3.19)
According to Eqs. (3.17) (3.19), rli and r2i are local
maxima, and r3i and r4i are equal local minima of r. Since