Integration

數學小子

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 [a,b]. By considering the integral of the function [λf(x)+g(x)]² on [a,b], set up a quadratic inequality in the parameter λ. Hence show that
[∫(a->b)f(x)g(x)dx]²≤{∫(a->b)[f(x)]²dx}{∫(a->b)[g(x)]²dx}.
b) Let f(x) be a non-constant function with continuous derivative on [0,1] 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∈[0,1].
(ii) use (i) and (a) to show that
[f(x)]²≤x∫(0->1/2)[f'(t)]²dt         if x∈[0,1/2]
and
[f(x)]²≤(1-x)∫(1/2->1)[f'(t)]²dt       if x∈[1/2,1].
(iii) Use (ii) to show that
∫(0,1)[f(x)]²dx≤1/8 ∫(0->1)[f'(x)]²dx.

-終場ソ使者-

Since,

by integration,

ie


b)i)
               (Since f(0)=0)
          (Since f(1)=0)
ii)
     (Since )



    (Since)


iii)諗多陣先