how to find the minimum value of $f(x)=\frac{1}{x\cos(x)}$ how to find the minimum value of $f(x)=\frac{1}{x\cos(x)}$? $x\in(0,\frac{\pi}{2})$
My try:
differential $f$:
$$\frac{df}{dx}=\frac{x\sin(x)-\cos(x)}{x^2\cos^2(x)}=0$$
then $x\tan(x)=1$. How to find the value of $x$?
I appreciate also other methods to find the minimum of $f(x)$.
 A: There is no closed solution for what you are trying to calculate. The minimum value can be calculated numerically, but you cannot get a "nice looking" solution, unfortunatelly.
A: This isn't an analytic solution, but here's how you'd find the solution numerically in $Mathematica$:
{#, NMinimize[
    {1/(x*Cos[x]), 0 < x < Pi/2},
    x,
    Method -> #
    ]} & /@ {"NelderMead", "DifferentialEvolution", "SimulatedAnnealing", "RandomSearch"}

The output is
{{"NelderMead", {1.78223, {x -> 0.860334}}},
 {"DifferentialEvolution", {1.78223, {x -> 0.860334}}},
 {"SimulatedAnnealing", {1.78223, {x -> 0.860334}}},
 {"RandomSearch", {1.78223, {x -> 0.860334}}}}

That is, all of the minimization methods found the same solution. For those unfamiliar with the $Mathematica$ language but wishing to manipulate the code, a simplified example is
NMinimize[{1/(x*Cos[x]), 0 < x < Pi/2}, x, Method -> "DifferentialEvolution"]

which outputs
{1.78223, {x -> 0.860334}}

A: Let us say that the function goes through an extremum for the value $a$ which is the solution of $a \tan(a)=1$. At this point, the value of the function is then $\sec (a) \tan (a)$. 
Now, the problem is to solve, using numerical methods, the equation $f(x)=x \tan(x)-1=0$. A classical method is Newton provided a reasonable starting point. By inspection, you would see that the function is negative at $\frac {\pi}{4}$ and positive at $\frac {\pi}{3}$. for this function Newton iterates will be given by $$\frac{2 x^2+\cos (2 x)+1}{2 x+\sin (2 x)}$$ Starting iterations at $x=\frac {\pi}{4}$, Newton iterates will successively be $0.868875$, $0.860441$, $0.860334$ which is the final solution.  
So the value of the function at the extremum is $\sec (0.860334) \tan (0.860334)=1.78223$. 
A: Best bet would be to try some Taylor polynomial approximation and then find the roots of the corresponding polynomial. 
