"Show that the following equation has exactly one real root:
$3x+2\cos x+5=0$"
So I think the way to approach this problem is to use a combination of the intermediate value theorem and the mean value theorem.
But since we are not given a specific interval, I'm not entirely sure what values of $f(a)$ and $f(b)$ to choose. Am I suppose to choose $f(0)$ and $f(1)$ and see that $0$ lies in between them. If so, how exactly can we proceed with the mean value theorem after using the intermediate value theorem. Do we have to check for continuity and the number $c$, or is there some other way?
Any help?
Best Answer
Let $f(x)=3x+2\cos x+5$. Then $f'(x)=3-2\sin x>0$. Thus $f$ is strictly increasing. Then graph of $f$ can intersect the $x$-axis at most once.
Note $f$ is continuous and $\lim_{x\rightarrow-\infty}f(x)=-\infty$ and $f(0)=5>0$, hence it's graph must intersect the $x$-axis at least once.