Use the Intermediate Value Theorem to show that $\cos(x)=x^3$ has a solution.

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I am not sure if I am fully understanding how to solve this, but I think that, since the since $g(x)=\cos(x)$ and $g(x)=x^3$ are continuous everywhere, the function $f(x)=\cos(x)-x^3$ must also be continuous everywhere, and therefore, according to the Intermediate Value Theorem, $\cos(x)=x^3$ must have a solution. However, I'm not sure if that's true.

How can I show that $\cos(x)=x^3$ has a solution?

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1 Answer

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Because $f(x)$ is continuous and hence satisfies the IVT, AND it is negative at say $x=\pi$, AND it is positive at say $x=0$, we know that between those two x values $f(x)=0$.

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