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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?

Ángel Mario Gallegos
  • 19,882
Víðir Helgi Helgason
  • 1
  • Hint: look at what happens as $x\rightarrow ±\infty$ – lulu Sep 24 '15 at 11:18
  • Further to lulu's hint and Mario G's answer, some things to think about are the continuity of the functions over their domains of definition (here all of the Reals). What are the ranges of $x^{179}$, of $1+x^2+sin^2x$, and so does the denominator go to zero, etc... – AmpleMimic Sep 24 '15 at 11:34
  • i think we get three Solutions for (iii) – Dr. Sonnhard Graubner Sep 24 '15 at 14:43

1 Answers1

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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 $f(x)=0$.

Ángel Mario Gallegos
  • 19,882