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.