If (m ) and (n ) are the least and greatest values of (|z| ) respectively and (|z-4+3 i| 1 ). Let (k ) be the least value of ( x^4+x^2+4 x ) on the interval ((0, ) ). Then (k= )

Step-by-step solution

( aligned |z-4+3 i| & 1 |Z-(4-3 i)| & 1 aligned ) It represent a circle of radius less than or equal to 1 unit Minimum value of (|z|=O P=O C-C P=5-1 ) (m=4 ) Maximum value of (|z|=O Q=O C+C Q=5+1 ) ( aligned n & =6 x^4+x^2+4 x & =x^3+x+ 4 x x^4+x^2+4 x & =x^3+x+ 1 x + 1 x + 1 x + 1 x aligned ) Since, ( AM GM ) ( aligned x^3+x+ 1 x + 1 x + 1 x + 1 x 6 & [6] x^3 x 1 x 1 x 1 x 1 x [6] 1 x^3+x+ 1 x + 1 x + 1 x + 1 x & 6 x^4+x^2+4 x 4 & 6 aligned ) ( ) Minimum Value of ( x^4+x^2+x 4 =6 ) ( array rr k=6 & [given] k=n & [