If the side length of square ABCD is 1 and point P moves on line AC, then the maximum value of AP & nbsp; · (Pb + PD) is___ ．

If the side length of square ABCD is 1 and point P moves on line AC, then the maximum value of AP & nbsp; · (Pb + PD) is___ ．

Taking the line where AB is located as the x-axis and the line where ad is located as the y-axis to establish the plane rectangular coordinate system, then a (0,0), B (1,0), C (1,1), D (0,1), AP = (m, m), Pb = (1-m, - M), PD = (- m, 1-m), & nbsp; AP · (Pb + PD) = (m, m) (1-2m, 1-2m) = 2m

If the side length of square ABCD is 1 and point P moves on line AC, then AP (vector) times the maximum value of (Pb + PD) (vector)?

Let a (0,0) B (1,0) C (1,1) d (0,1)
P(x,x) x∈[0,1]
=(x,x)
=(1-x,-x)
=(-x,1-x)
(+)=(x,x)(1-2x,1-2x)=2x(1-2x)=-(2x-1/2)^2+1/4
When x = 1 / 4, the maximum is 1 / 4
When x = 0, the value is 0, when x = 1, the value is - 2, so the minimum value is - 2
So the value range is [- 2,1 / 4]

If the side length of square ABCD is 1 and point P moves on line AC, then the value range of AP & nbsp; · & nbsp; (Pb + PD) is______ ．

Let a (0,0), B (1,0), C (1,1), D (0,1) set P (x, x) (0 ≤ x ≤ 1) AP = (x, x), & nbsp; Pb = (1 − x, − x), PD = (− x, 1 − x) ｜ AP · & nbsp; (Pb + PD) = 2x (1-2x) = − 4 (x − 14) 2 + 14 & nbsp

If the side length of square ABCD is 1 and point P moves on line AC, then the value range of vector AB * (vector Pb + vector PD) is?

Let a (0,0) B (1,0) C (1,1) d (0,1)
P(x,x) x∈[0,1]
=(x,x)
=(1-x,-x)
=(-x,1-x)
(+)=(x,x)(1-2x,1-2x)=2x(1-2x)=-(2x-1/2)^2+1/4
When x = 1 / 4, the maximum is 1 / 4
When x = 0, the value is 0, when x = 1, the value is - 2, so the minimum value is - 2
So the value range is [- 2,1 / 4]