MathLabs

Problem 4

Show that the solution set of the inequality ∑k=170kx−k≥54\sum_{k=1}^{70}\frac{k}{x-k}\ge\frac{5}{4} is a union of disjoint intervals whose total length is 19881988.
Step 1 of 4: Define the rational function
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f(x)=∑k=170kx−kf(x)=\sum_{k=1}^{70}\frac{k}{x-k}
Detailed analysis

Let f(x)=∑k=170kx−kf(x)=\sum_{k=1}^{70}\frac{k}{x-k}. On every interval (n,n+1)(n,n+1) with 1≤n≤691\le n\le69, each summand is strictly decreasing where defined, so ff is strictly decreasing. As x→n+x\to n^+, the term n/(x−n)n/(x-n) tends to +∞+\infty; as x→(n+1)−x\to(n+1)^-, the term (n+1)/(x−n−1)(n+1)/(x-n-1) tends to −∞-\infty. Hence the intermediate value theorem gives a unique rn∈(n,n+1)r_n\in(n,n+1) with f(rn)=5/4f(r_n)=5/4.