I f t h e r a n g e o f a t h r e e v a r i a b l e f u n c t i o n f ( x , y , z ) = x 2 + y 2 + z 2 x y + y z + x z c a n b e w r i t t e n a s [ a , b ] . F i n d a + b .
Note: ( a , b , c ) = ( 0 , 0 , 0 ) . The values could be negative.
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That's a really nice interpretation with the dot product, relating it to the cosine of the angle between these vectors.
W e k n o w t h a t : ( x + y + z ) 2 = ( x 2 + y 2 + z 2 + 2 x y + 2 y z + 2 x z ) ≥ 0 ⇒ x 2 + y 2 + z 2 x y + y z + x z ≥ − 2 1 . . . . . . . . . . . ( i ) N o w : x 2 + y 2 + z 2 − x y − y z − x z = 2 1 ( ( x − y ) 2 + ( y − z ) 2 + ( x − z ) 2 ) ≥ 0 ⇒ x 2 + y 2 + z 2 x y + y z + x z ≤ 1 . . . . . . . . . . . . . ( i i ) B y ( i ) a n d ( i i ) w e c a n s a y t h a t r a n g e o f f ( x , y , z ) i s [ 2 − 1 , 1 ] ∴ a + b = 0 . 5
SInce ( x + y + z ) 2 = ( x 2 + y 2 + z 2 ) + 2 ( x y + y z + z x ) , the given function is equal to f ( x , y , z ) = 2 1 ( x 2 + y + 2 + z 2 ( x + y + z ) 2 − 1 ) . Interpreting R 2 = x 2 + y 2 + z 2 as the square radius of a sphere centered at zero, and x + y + z as the dot product of a vector lying on that sphere with 1 , 1 , 1 .
Maximum value: Choose vector on the sphere parallel to 1 , 1 , 1 . The coordinates are x = y = z = ± 3 1 3 R and we find ∣ x + y + z ∣ = 3 R .
Minimum value: Choose vectors on the sphere perpendicular to 1 , 1 , 1 . There are infinitely many such vectors (lying on a ring). They satisfy ∣ x + y + z ∣ = 0 .
Substituting the minimum and maximum values of ∣ x + y + z ∣ into the equation, we find that f lies between 2 1 ( ( 3 ) 2 − 1 ) = 1 (max) and 2 1 ( 0 2 − 1 ) = − 2 1 (min).
Sorry but i think that this question is overrated. It should be of level 3
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For the sake of variety, let me submit a geometrical solution; I like to see what I am doing.
Using the dot product, we can write f ( x , y , z ) = ∣ ∣ ( x , y , z ) ∣ ∣ 2 ( x , y , z ) ⋅ ( y , z , x ) = cos ( θ ) , where θ is the angle between the vectors ( x , y , z ) and ( y , z , x ) . Now ( y , z , x ) is obtained by rotating ( x , y , z ) through 2 π / 3 about the line spanned by ( 1 , 1 , 1 ) , a permutation of the axes. Thus 0 ≤ θ ≤ 2 π / 3 and [ a , b ] = [ cos ( 2 π / 3 ) , cos ( 0 ) ] = [ − 1 / 2 , 1 ] . Finally, a + b = 0 . 5 .