The function satisfies f( 1+t)=f( 1-t), that is, its symmetry axis is x= 1.
∴-b/2= 1 = & gt； b=-2
f(x)=x^2+bx+c=x^2-2x+c=(x- 1)^2+c- 1
The minimum value is c- 1=4 = >c=5.
The resolution function is y = x 2-2x+5.
g(x)=f(sinx)-cos？ x-msinx+6
= sin? x-2sinx+5-msinx+6
= sin? x-(2+m)sinx+ 1 1
=[sinx-(2+m)/2]^2+ 1 1-(2+m)^2/4
=[u-(2+m)/2]^2+ 1 1-(2+m)^2/4
=s(u)
The function s(u) is an upward parabola with u=sinx as a variable, and its value is the same as g(x).
The definition domain of this parabola is u∈[- 1, 1], and the symmetry axis is u=(2+m)/2.
Need to discuss the value of m:
① If the symmetry axis u = (2+m)/2
The function is monotone increasing function on [- 1, 1], and the minimum value is s(- 1)=m+ 14=2.
The solution is m=- 12 and satisfies m < -4.
② if the symmetry axis u = (2+m)/2 >; 1, that is, m> So, 0
The function monotonically decreases in [- 1, 1], and the minimum value s( 1)=-m+ 10=2.
The solution is m=8, which satisfies m>0.
③ If the symmetry axis is-1≤(2+m)/2≤ 1, that is, -4≤m≤0, then
The function decreases first and then increases on [- 1, 1], and the minimum value is 1 1-(2+m) 2/4 = 2.
The solution is m=-8 or 4, which does not satisfy -4≤m≤0, so there is no solution for m at this time.
To sum up, when m=- 12 or 8, the minimum value of the function g(x) is 2.