Thread: Prove x^3 = cosx + 16

1. Prove x^3 = cosx + 16

Prove x^3 = cosx + 16

2. Originally Posted by johnlaw
Prove x^3 = cosx + 16

As is, this question makes no sense. Do you have any other information?

Prove that the equation x^3 = cosx + 16 has a solution

that should help.. sorry for the miswording

4. Originally Posted by johnlaw

Prove that the equation x^3 = cosx + 16 has a solution

that should help.. sorry for the miswording
That's better. You can prove there is a solution by graphing both $y_1 = x^3$ and $y_2 =\cos x + 16$ on the same set of axis. If they intersect you have proven there is a solution.

Finding that solution is a new world of pain.

5. Having graphed it, I see that there is an intersection at x slightly larger than 2. Using that, I can give a more formal proof by noting that $2^3= 8$ while $cos(2)+ 16$ is greater than 15.5 and that $3^3= 27$ while $cos(3)+ 16$ is only slightly larger than 15.

That is, $2^3< cos(2)+ 16$ while $3^3> cos(3)+ 16$. Since $x^3$ and cos(x)+ 16 are continuous functions, there must be x between 2 and 3 such that $x^3= cos(x)+ 16$.

6. Originally Posted by johnlaw