1, solution: (1) Because the symmetry axis of the quadratic function f (x) = ax 2+bx+a is x=7/4.

So -b/2a=7/4.

And the equation f(x)=7x+a has two equal real roots.

So the discriminant δ of equation f(x)=7x+a = (b-7) 2-4a * 0 = 0.

So b=7.

So a=-2

So f (x) =-2x 2+7x-2.

(2) The maximum value of f (x) is f (7/4) =-2 * (7/4) 2+7 * (7/4)-2 = 33/8.

The minimum value of f(x) is f (3) =-2 * 3 2+7 * 3-2 = 1.

So the range of f(x) is [1, 33/8].

(3) f(3)=65438+ 0 from (2)

If M=7/4, then 3/M= 12/7≠33/8, so it does not conform.

Then m > 7/4

Then f (m) =-2m 2+7m-2 = 3/m.

So 2m 3-7m 2+2m+3 = 0.

Only M=3 is consistent with the solution, and all other solutions are inconsistent.

So M=3

2、

Solution: ∫f(x)= 4x+m×2x+ 1.

=(2^x)^2+m×2^x+ 1

If f(x) has one and only one zero.

That is, the equation (2 x) 2+m× 2 x+ 1 = 0 has one and only one real root.

Let t = 2 x and t > 0.

That is, the equation T 2+MT+ 1 = 0 has one and only one real root in (0, +∞).

Let g (t) = t 2+mt+ 1.

∴△ = m 2-4 = 0 or△ = m 2-4 > 0 or△ = m 2-4 > 0.

-m/2>0 g (0)0

∴m=-2

The above is from Baidu.