Prove that there is a solution

Question

Prove that there is a solution for some $x$, $$x^{179}+\frac{163}{1+x^{2}+\sin^{2}(x)}=119$$

Answer 1

We have the equation :

$$x^{179}+\frac{163}{1+x^{2}+\sin^{2}(x)}=119$$

Let's consider the function $f$, such that $f : \mathbb R \to \mathbb R$

$$f(x) = x^{179}+\frac{163}{1+x^{2}+\sin^{2}(x)}-119$$

The function $f$ is continuous in $\mathbb R$ and also :

$$f(0) = 163-119 >0$$

$$f(1) = 1 + \frac{163}{2+\sin^2(1)}-119<0$$

since we have that $\sin^2(1)>0$ and then $\frac{163}{2+\sin^2(1)}<\frac{163}{2}$.

Since $f$ is continuous over $\mathbb R$, it will also be continuous in the interval $[0,1]$ which means that by Bolzano's Theorem, $\exists \space x_0 \space \in (0,1) : f(x_0) = 0$, which means that there exists a solution for the initial expression.

Answer 2

HINT

Show that

$$f(x) =x^{179}+\frac{163}{1+x^{2}+\sin^{2}(x)}$$

is surjective (use limits and continuity) then refer to IVT.