It is known that the quadratic function f (x) satisfies f (x + 1) - f (x) = 2x (x ∈ R), and f (0) = 1 It is known that f (x) is a quadratic function. For any x belonging to R, f (x + 1) - f (x) = - 2x + 1 and f (0) = 1 are satisfied (1) Finding the analytic expression of F (x) (2) When x belongs to [- 2,1], the image of y = f (x) is always above the image of F = - x + m, and the value range of real number m is obtained I would like to ask the following constant set up into a solution Is m ∈ [- 1,5] the answer? (2) In the question, f (x) = 2x + m has a solution, and the range of real number m is obtained

It is known that the quadratic function f (x) satisfies f (x + 1) - f (x) = 2x (x ∈ R), and f (0) = 1 It is known that f (x) is a quadratic function. For any x belonging to R, f (x + 1) - f (x) = - 2x + 1 and f (0) = 1 are satisfied (1) Finding the analytic expression of F (x) (2) When x belongs to [- 2,1], the image of y = f (x) is always above the image of F = - x + m, and the value range of real number m is obtained I would like to ask the following constant set up into a solution Is m ∈ [- 1,5] the answer? (2) In the question, f (x) = 2x + m has a solution, and the range of real number m is obtained

f(0)=1
Let f (x) analytic formula F (x) = ax & # 178; + BX + 1
f(x+1) -f(x)=-2x+1
a(x+1)²+b(x+1)+1-ax²-bx-1
=2ax+(a+b)=-2x+1
∴2a=-2
a+b=1
∴a=-1
b=2
Analytic formula F (x) = - X & # 178; + 2x + 1
(2)
When x belongs to (- 2,1),
F (x) is above the image of y = - x + M
∴-x²+2x+1>-x+m
-x²+3x+1>m
G (x) = - X & # 178; + 3x + 1 the axis of symmetry is x = 3 / 2
The minimum value of G (x) = g (- 2) = - 4-12 + 1 = - 15 > m
m