$\min$ value of $f(x)=a\sec x+b\csc x\;,$ Where $a,b>0$ and $\displaystyle x\in \left(0,\frac{\pi}{2}\right)$

Question

$\min$ value of $f(x)=a\sec x+b\csc x\;,$ Where $a,b>0$ and $\displaystyle x\in \left(0,\frac{\pi}{2}\right)$

Although we can solve it using Derivative Test, But my question is can we solve it using $\bf{cauchy}$

or Using $\bf{A.M\geq G.M}$ or $\bf{Holder\; Inequality}$

Help required, Thanks

Answer 1

Let $p=3/2,q=3$ then $1/p+1/q=1$. Now by Hölder's inequality, $$(((a\sec x)^\frac1p)^p+((b\csc x)^\frac1p)^p)^\frac1p(((\cos^2)^\frac1q)^q+((\sin^2 x)^\frac1q)^q)^\frac1q\ge (a\sec x)^\frac1p(\cos^2)^\frac1q)+(b\csc x)^\frac1p)((\sin^2 x)^\frac1q)=a^\frac23+b^\frac23. $$

Indeed there's another approach using physical method (static electrical energy in particular). I'll add it to the answer if I have time.