Finding function expression Let f (x) = AX2 + BX + C pass through points (0,1) and (1,4), and the inequality f (x) ≥ 4x holds for any real number X ① Find the expression of function f (x); ② Let g (x) = KX + 1, if f (x) = log2 (g (x) - f (x)) is an increasing function in the interval [1,2], find the value range of real number K

Finding function expression Let f (x) = AX2 + BX + C pass through points (0,1) and (1,4), and the inequality f (x) ≥ 4x holds for any real number X ① Find the expression of function f (x); ② Let g (x) = KX + 1, if f (x) = log2 (g (x) - f (x)) is an increasing function in the interval [1,2], find the value range of real number K

① First, since it is determined to be a quadratic function, a ≠ 0
Substituting points (0,1) and (1,4) into f (x) = ax & sup2; + BX + C, C = 1,4 = a + B + C
We can get: B = 3-A, C = 1, substituting into the expression of F (x)
f(x)=ax²+(3-a)x+1
From F (x) ≥ 4x we get ax & sup2; + (3-A) x + 1 ≥ 4x
ax²-(1+a)x+1≥0
According to the above formula, it holds for any real number x, that is, the image of the function H (x) = ax & sup2; - (1 + a) x + 1 is always on the x-axis or on the x-axis
So there must be:
a>0
△=(1+a)²-4a≤0
If the latter formula is reduced to (1-A) & sup2; ≤ 0, then it can only be (1-A) & sup2; = 0
The simultaneous solution of the two equations is a = 1, and then f (x) = ax & sup2; + (3-A) x + 1
f(x)=x²+2x+1
② F (x) = log2 (g (x) - f (x)) = log2 [(KX + 1) - (X & sup2; + 2x + 1)] = log2 [- X & sup2; + (K-2) x] = = let = = log2 [u (x)]
It can be seen that f (x) is a compound function, because f (x) is an increasing function in the interval [1,2], and log2 [u (x)] is an increasing function of U (x)
U (x) = - X & sup2; + (K-2) x is an increasing function in the interval [1,2], and U (x) is a new quadratic function with the opening downward and the axis of symmetry x = K-2
k-2≥2 ……………… one
If the combined true number is greater than 0, u (1) > 0 and U (2) > 0, i.e
-4+2(k-2)>-1+(k-2)>0 ……………… two
The range of the real number k is obtained from the simultaneous solution of one or two inequalities
k>5