Question

For the function f(x)= x^2-2 defined over [1,5] , verify: L(P,f)≤U(-P,f) where P is the partition which divides [ 1,5] into four equal intervals.

Answer 1

$L(P,f)= \sum^4_{i=1}m_i \Delta x_i\\ U(-P,f)=\sum^4_{i=1}M_i \Delta x_i\\ m_i=infimum[f(x):x_{i-1}≤x≤x_i]\\ M_i=inprenum[f(x):x_{i-1}≤x≤x_i]\\ f(1)=-1 \\ f(2)=2 \\ f(3)=7\\ f(4)=14 \\ f(5)=23 \\ Hence, m_1=-1, m_2=2, m_3=7, m_4=14\\ M_1=2, M_2=7, M_3=14, M_4=23\\ \Delta x_i=1\\ L(P,f)=-1+2+7+14=22\\ U(-P,f)=2+7+14+23=46\\ L(P,f)≤U(-P,f)\\$