When the function y = - X3 + 3x + A is used to find the value of a, the equation y = 0 has two real roots

When the function y = - X3 + 3x + A is used to find the value of a, the equation y = 0 has two real roots

Y '= - 3x ^ 2 + 3 = 0, the extreme point x = 1, - 1
F (1) = 2 + A is the maximum,
F (- 1) = - 2 + A is the minimum
If there are two real roots, then one of them must be a multiple root, which means that the maximum or minimum is tangent to the x-axis. Moreover, the maximum must be non negative and the minimum non positive
f(1)=2+a>=0 ==> a>=-2
f(-1)=-2+a a

If the moving line x = a intersects the image of F (x) = SiN x and G (x) = cos x at M and N points respectively, then the maximum value of | Mn | is
I feel the answer is wrong
I read it wrong. The answer is right

The intersection points of x = A and f (x) = SiN x and G (x) = cos X are m (a, Sina) and n (a, COSA) respectively
|MN|=|sina-cosa|
|Mn | ^ 2 = (Sina COSA) ^ 2 = 1-2sinacosa = 1-sin2a, when a = k π - π / 4, take the maximum value of 2
Because | Mn | > = 0, when a = 0, | Mn | also has a maximum √ 2

If the moving line x = A and the function f (x) = 3sin (x + π 6) and G (x) = cos (x + π 6) intersect at two points m and N respectively, then the maximum value of | Mn | is______ ．

When x = a, | Mn | = | f (a) - G (a) | = | 3sin (a + π 6) - cos (a + π 6) = | 2Sin (a + π 6 - π 6) | = 2 | Sina |, when | Sina | = 1, | Mn | achieves the maximum value of 2, so the answer is: 2