I've the following optimization problem:$$\min f(\theta_1,\theta_2)=\frac{a}{\cos\theta_1\cdot v_1}+\frac{b}{\cos\theta_2\cdot v_2}$$$$\operatorname{sub}\quad a\cdot\tan\theta_1+b\cdot\tan\theta_2=c$$ where $a,b,c,v_1,v_2$ are positive constants. Once solved it gives the following identity, known as the Snell's law of refraction:$$\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}$$My question is: is there a way to show that if the above equation is satisfied then $f$ is minimized?
Remember that when you solve the system of partial derivatives of the Lagrangian function you get critical points; so how could I show that $f$ has a global minimum?
solove the equation $a\tan(\theta_1)+b\tan(\theta_2)=c$ for $\theta_2$ and plug this in your function and you will get a problem in only one variable.