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[ 東京工業大学 1960 年 数学Ⅲ ２ ]

a≦ ≦x b (ab) で f x( )0 のとき， 1 2

( ) , ( ) ( ) ( )

b b

a a

N f x dx v t x t f x dx

N とおく。

( )

v t を最小にするtの値をM とする。

(1) M を求めよ。

(2) v t( )v M( ) (t M)2 であることを示せ。

(1) 1 2

( ) b( ) ( )

v t a x t f x dx

N

2 2

1 b( 2 ) ( )

a x tx t f x dx

N

2 2

1 b ( ) 2 b ( ) b ( )

a x f x dx t a xf x dx t a f x dx

N

2 2

1 2

( ) ( )

b b

a a

x f x dx t xf x dx t

N N

2 2

1 1 1 2

( ) ( ) ( )

b b b

a a a

t xf x dx xf x dx x f x dx

N N N

 

…①

よって，v t( ) を最小にする t は 1

b ( ) M a xf x dx

N である。

(2)

2

1 1 2

( ) b ( ) b ( )

a a

v M xf x dx x f x dx

N N

 

であるから

①より

1 2

( ) b ( ) ( )

v t t a xf x dx v M N

 

なので

1 2

( ) ( ) b ( )

v t v M t a xf x dx N

 

(t M)2

  となる。

参照

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