Tau(q) exercise solutions

Pictured here are some points on the tau(q) graph for these values: r = 0.5, p1 = 0.05, p2 = 0.2, p3 = 0.3, and p4 = 0.45.
Inspection of the graph certainly suggests that the graph of tau(q) is decreasing.
 
Recall the the calculation
      dtau/dq = -( piqritau(q)(ln(pi)) / ( piqritau(q)(ln(ri))
In the special case that all ri = r, this simplifies to
      dtau/dq = -( piq(ln(pi)) / (ln(r) piq)
Because 0 < r < 1, ln(r) < 0 and so the denominator is negative.
Because each 0 < pi < 1, ln(pi) < 0 and so each term of the numerator is negative.
Then the minus in front of the fraction gives dtau/dq < 0. That is, the graph of tau(q) is decreasing.
The general case of distinct ri follows by noting that each 0 < ri < 1, so each ritau(q) > 0 and each ln(ri) < 0. Then each term of the denominator is negative and each term of the numerator is negative.

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