The thermal resilience index, RRR, of a specific polymer during a stress test is modeled by the function
R(t)=5t2−202t2+9t+10−22t+5,t∈R, t>1 R(t) = \frac{5t^2 - 20}{2t^2 + 9t + 10} - \frac{2}{2t + 5}, \quad t \in \mathbb{R}, \ t > 1 R(t)=2t2+9t+105t2−20−2t+52,t∈R, t>1where ttt is the duration of the test in hours.
Show that R(t)=5t−122t+5R(t) = \frac{5t - 12}{2t + 5}R(t)=2t+55t−12.
Show, using calculus, that RRR is an increasing function for all t>1t > 1t>1. You must make your reasoning clear.
The monitoring function HHH is defined by
H(t)=8+2lnt,t≥1 H(t) = 8 + 2 \ln t, \quad t \ge 1 H(t)=8+2lnt,t≥1Find H−1(x)H^{-1}(x)H−1(x).
Find the exact value of aaa for which HR(a)=9HR(a) = 9HR(a)=9.
