I am trying to calculate the maximum and minimum points (between $-3\pi$ and $3\pi$) of $$f(x)=\frac{\sin(x)}{x}$$
I have found the derivative of the function and let it equal to zero.
$$f'(x)=\frac{x\cos(x) – \sin(x)}{x^2}$$
$$f'(x)=0$$
$$\frac{x\cos(x) – \sin(x)}{x^2}=0$$
$$x\cos(x) – \sin(x)=0$$
$$x\cos(x)=\sin(x)$$
$$x=\tan(x)$$
I am unaware as to how to find $x$. I assume that once I find $x$, I can use a sign diagram or second derivative test to determine the minimum and maximum values. Any help would be highly appreciated.
Best Answer
Equations which contains polynomial and trigonometric functions do not show explicit solutions and numerical methods should be used to find the roots.
The simplest root finding method is Newton; starting from a reasonable guess $x_0$, the method will update it according to $$x_{n+1}=x_n-\frac{F(x_n)}{F'(x_n)}$$ So, in your case, $$F(x)=x \cos(x)-\sin(x)$$ I think it is better to let it under this form because of the discontinuities of $\tan(x)$.
You can notice that if $x=a$ is a root, $x=-a$ will be another root. So, let us just focus on $0\leq x \leq 3\pi$. If you plot the function, you notice that, beside the trivial $x=0$, there are two roots located close to $5$ and $8$. These would be the guesses.
Using $F'(x)=-x \sin (x)$, the iterative scheme then write $$x_{n+1}=x_n-\frac{1}{x_n}+\cot (x_n)$$ Let us start with $x_0=5$; the method then produces the following iterates : $4.50419$, $4.49343$, $4.49341$ which is the solution for six significant figures.
I let you doing the work for the other solution.