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Senior one finishes mathematics in Dalian.
According to the symmetry of quadratic function, the symmetry axis of the function is x=-b/2.

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.