Given that f (x) = x ^ 3 + 2x, the inequality f (cos2a-3) + F (2m Sina) > 0 holds for any a ∈ R, and the value range of real number m is obtained

Given that f (x) = x ^ 3 + 2x, the inequality f (cos2a-3) + F (2m Sina) > 0 holds for any a ∈ R, and the value range of real number m is obtained

It is easy to get that: F (x) is an odd function, and f (x) is increasing on R. the original inequality is: F (cos2a-3) > - f (2m Sina) because f (x) is an odd function, so, - f (2m Sina) = f (sina-2m) inequality is: F (cos2a-3) > F (sina-2m) because f (x) is an increasing function, so: cos2a-3 > sina-2m2m > - cos2a + Sina + 3

[with addition] if the inequality | x + 3 | + | X-1 | > A is constant, the value range of a is obtained
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If the inequality | x + 3 | + | X-1 | > a holds
Only a is less than the minimum value of | x + 3 | + | X-1 |
|X + 3 | + | X-1 | is the sum of the distances from any point on the number axis to - 3 and 1
Obviously, when x is between - 3 and 1, the sum of the distances to the two points is the shortest, which is 4
That is to say, the minimum value of | x + 3 | + | X-1 | is 4
So the value range of a is a