Prove that the equation $$\ x^4 = 18 + \frac{1}{1-x} $$ has at least three roots.
How do I use the intermediate value theorem to prove this? Does it mean I just need to find three different values for which $\ f(x) = x^4 – 18 – \frac{1}{1-x} $ is positive and therefore it will have a root?
Best Answer
Let $$g(x)= x^5-x^4-18x+19$$ Since we are looking for solution of $g(x)=0$ and we have $g(-3)<0$, $g(0)>0$, $g(2)<0$ and $g(3)>0$ we are done.