Question 8.4: Repeat Example 8.3 without differentiating.

Using Eq. (8.37),

\sum\limits_{1}^{m} \frac{1}{σ  +  z_{i} } = \sum\limits_{1}^{n} \frac{1}{σ  +  p_{i} }     (8.37)

\frac{1}{σ  −  3}  +  \frac{1}{σ  −  5} = \frac{1}{σ  +  1}  +  \frac{1}{σ  +  2}                             (8.38)

Simplifying,

11σ²  −  26σ  −  61 = 0                (8.39)

Hence, σ = − 1.45 and 3.82, which agrees with Example 8.3 .