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Question Number 11299 by agni5 last updated on 19/Mar/17
![Define two partition p_1 and p_2 of [2,5] such that p_1 ⊂p_2 . Find the upper and lower product sums with respet to f ,?defined by f(x)=x , x<4 =1−x^2 , x ≥4 . Also verify the relationship between these 4 sums.](Q11299.png)
$$\mathrm{Define}\:\mathrm{two}\:\mathrm{partition}\:\mathrm{p}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{p}_{\mathrm{2}} \:\mathrm{of}\:\left[\mathrm{2},\mathrm{5}\right]\:\mathrm{such}\:\mathrm{that} \\ $$ $$\mathrm{p}_{\mathrm{1}} \subset\mathrm{p}_{\mathrm{2}} \:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{upper}\:\mathrm{and}\:\mathrm{lower}\:\mathrm{product} \\ $$ $$\mathrm{sums}\:\mathrm{with}\:\mathrm{respet}\:\mathrm{to}\:\mathrm{f}\:,?\mathrm{defined}\:\mathrm{by} \\ $$ $$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}<\mathrm{4} \\ $$ $$\:\:\:\:\:\:\:\:=\mathrm{1}−\mathrm{x}^{\mathrm{2}} \:,\:\:\:\:\:\mathrm{x}\:\geqslant\mathrm{4}\:\:. \\ $$ $$\mathrm{Also}\:\mathrm{verify}\:\mathrm{the}\:\mathrm{relationship}\:\mathrm{between}\:\mathrm{these} \\ $$ $$\mathrm{4}\:\mathrm{sums}. \\ $$