Let f(x)= ( a^ ^4 x a^ 2 ^2 x )^3 (a^ 5 ^3 x a^ 5 x ) and b=( 5 )^ 2 ₅ 3 (a 0, a 1) . If the minimum value of (f(x) a^b )^ cosec ^2 x + _a x is equal to a^k, k N , find k .

Step-by-step solution

f(x)=a^ 3 ^4 x+5 ^3 x-6 ^2 x+5 x and b=3 ( x 0) (f(x) a^3 )^ cosec ^2 x =a^ 3 ^2 x+5 x-6+ 5 x + 3 ^2 x = a ^ 3 ( ^2 x+ 1 ^2 x )+5 ( x+ 1 x )-6 min. value of 3 ( ^2 x+ 1 ^2 x )+5 ( x+ 1 x )-6 ( for x 0)=3(2)+5(2)-6=10 ( x 0 and the value is minimum at x =1 ) _a( x)=0