Integration
素聞呢題係HKALE既試題,不過搵極都搵唔到係邊年a) Let f(x) and g(x) be two functions continuous on the interval . By considering the integral of the function [λf(x)+g(x)]2 on , set up a quadratic inequality in the parameter λ. Hence show that
[∫(a->b)f(x)g(x)dx]2≤{∫(a->b)2dx}{∫(a->b)2dx}.
b) Let f(x) be a non-constant function with continuous derivative on satifying f(0)=0 and f(1)=0.
(i) show that
f(x)=∫(0->x)f'(t)dt=-∫(x->1)f'(t)dt
for any x∈.
(ii) use (i) and (a) to show that
2≤x∫(0->1/2)2dt if x∈
and
2≤(1-x)∫(1/2->1)2dt if x∈.
(iii) Use (ii) to show that
∫(0,1)2dx≤1/8 ∫(0->1)2dx. 本帖最後由 -終場ソ使者- 於 5-4-2011 16:14 編輯
Sincehttp://upload.wikimedia.org/math/7/4/0/740601c8eaa10009780a1db728a217ee.png,
http://upload.wikimedia.org/math/d/8/f/d8fc033965d44ffc9d6645968ebde63c.png
by integration,
http://upload.wikimedia.org/math/a/b/b/abba13d25766f72afed7bc20cced8982.png
ie http://upload.wikimedia.org/math/5/b/c/5bc905767fa823af270f963211c2adbf.png
http://upload.wikimedia.org/math/a/2/5/a25fc5e86b5e4cd352bd9ce48fc553f1.png
http://upload.wikimedia.org/math/4/d/e/4de24e67add60e7e8432a4eb11d84e37.png
b)i)
http://upload.wikimedia.org/math/5/b/c/5bc53f8a040ab771b2fc74983c08e833.png (Since f(0)=0)
http://upload.wikimedia.org/math/0/a/e/0ae427b0ee3504eb0aa3cc0feeebf828.png (Since f(1)=0)
ii)
http://upload.wikimedia.org/math/6/0/5/60544325c8c86919ab52addc25af68b0.png (Since http://upload.wikimedia.org/math/5/6/e/56e1b05b30334895b144277795013a1b.png)
http://upload.wikimedia.org/math/3/0/f/30f751129bdadcf1df52e3bb7265b1ba.png
http://upload.wikimedia.org/math/3/5/a/35af6963b66836ac675f74ec67512bf6.png (Sincehttp://upload.wikimedia.org/math/b/4/3/b43fced2509828edd4e7f67190daf531.png)
http://upload.wikimedia.org/math/a/d/e/ade64caa5e6248401f1fa485c99509e5.png
iii)諗多陣先
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