Question Number 118084 by bemath last updated on 15/Oct/20
![suppose f: [1,3 ]→ [−1,1 ] such that ∫_1 ^3 f(x) dx = 0 . What the maximum value of ∫_1 ^3 x^(−1) .f(x) dx ?](Q118084.png)
$$\mathrm{suppose}\:\mathrm{f}:\:\left[\mathrm{1},\mathrm{3}\:\right]\rightarrow\:\left[−\mathrm{1},\mathrm{1}\:\right]\:\mathrm{such} \\ $$$$\mathrm{that}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{0}\:.\:\mathrm{What}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\:\mathrm{x}^{−\mathrm{1}} .\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$
Commented by john santu last updated on 16/Oct/20
$${max}\:=\:\mathrm{2ln}\:\mathrm{2}−\mathrm{ln}\:\mathrm{3} \\ $$