Under which one of the following conditions does the function f(x)=(p x)^2+(q cosec x)^2 attain minimum value?

Step-by-step solution

aligned & f(x)=(P x)^2+(q cosec x)^2 & f^ (x)=2 P^2 ^2 x x-2 q^2 cosec ^2 x x=0 & p^2 q^2 = cosec ^2 x x ^2 x x = ^4 x & ^2 x= p q ^2 x= q p & f^ (x)=2 p^2 ( x+ ^3 x )-2 q^2 ( x+ ^3 x ) & f^ (x)=2 p^2 ^2 x (1+3 ^2 x )+2 q^2 cosec ^2 x & (1+3 ^2 x ) & For ^2 x= q p & f^ (x) 0 & f(x) is minimum when ^2 x= q p aligned