Differentiating |
 | piqriβ(q) = 1 |
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twice with respect to q gives |
 |
piqriβ(q)((d2β/dq2)ln(ri) +
(ln(pi) + ln(ri) dβ/dq)2) = 0   (*) |
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Solving for d2β/dq2 we find |
d2β/dq2 = -( |  |
piqriβ(q)(ln(pi) + (dβ/dq)ln(ri))2)/( |
 |
piqriβ(q)(ln(ri))) |
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Because each piqriβ(q) > 0
and each ln(ri) < 0, we see d2β/dq2 >= 0. |
Finally, suppose d2β/dq2 = 0. Then equation (*) becomes |
 |
piqriβ(q)( ln(pi) +
ln(ri) dβ/dq)2 = 0 |
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Consequently, for each i, dβ/dq = -ln(pi)/ln(ri).
That is, all the ln(pi)/ln(ri) are equal. |
That is, if not all the ln(pi)/ln(ri) are equal,
then d2β/dq2 > 0. |
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