Question

Prove that?

`arcsin(x)^3 + arccos(x)^3 = alpha x` has no roots for `alpha<1/32` and `alpha>7/8`

Answer 1

I don't think it's true.

Explanation:

See this example%5E3+%2B+arccos(x)%5E3,+10x), in which I plot `arcsin^3(x)+arccos^3(x)` and `10x`.

According to your claim, they should have no intersection, since `alpha=10>7/8`, but as you can see, they have.