Let a function f: R R be defined as f(x)= [ array ll x, & a b array . If f(x) is derivable x R , find the minimum value of |[a+b+c+d]| . [Note: ['] denotes the greatest integer function.]

Step-by-step solution

f(x)= [ array ll x, & a < x b c-x-d, & x (- , a] x-c-d, & x (b, ) array . [Note: C must lies between a and b .] Now, aligned & . array l f^ (a⁺ )= a f^ (a⁻ )=-1 array ] a=-1 a=(2 n+1) , n I & . array l f^ (b⁻ )= b f^ (b⁺ )=1 array ] b=1 b=2 m , m I & aligned array ll & a=0= b & f(a)=0=f(b) array and Coordinate f (a⁺ )=f (a⁻ ) 0=c-a-d f (b⁺ )=f (b⁻ ) b-c-d=0 b=c+d |[a+b+c+d]|=|[a+2 b]|=[(2 n+1) +4 m ]=|[(2 n+1+4 m) ]|_ min. =4 Which occur at n=-1 and m=0 since a < b (note this point).