推 boxgene:S(1-x)f(x^2)dx <=怎麼知道他大於零....謝謝 04/12 20:40
推 pboywc:x在0和1之間 f(x)恆正由題設 04/12 23:28
推 boxgene:(1-x)不會有影響嗎 04/13 00:54
※ 引述《boxgene (原來什麼事情都不簡單)》之銘言：
: Prove the following statement :Let f be a positive continuous function
: on [0 ，1] . Suppose that
: 1 1
: S f(x) dx=a and S f(x^2) dx =b
: 0 0
: Then a<２b
: ¯
: 請問這要怎麼做
1 1
令x=t^2 a=S 2tf(t^2)dt =2S xf(x^2)dx
0 0
1 1
2b-a=2S f(x^2)dx - 2S xf(x^2)dx
0 0
1
2S (1-x)f(x^2)dx > 0
0
a<2b
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