Spivak chapet 7 problem 3. 4th edition, can someone help?

Question

Prove that there is some number $x$ such that

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

(ii) $\sin x = x - 1$

I am not sure what to do, can someone show me what to do?

Answer 1

For (ii) let $f:[0,3]\to\mathbb{R}$ defined by $f(t)=t-\sin t-1$, this is a continuous function, since $f(0)=-1$ and $f(3)=2-\sin 3>0$ it follows, from the Intermediate Value Theorem, that there is $x\in[0,3]$ such that $f(x)=0$.