Let f(x)= array cc 1+ x 2 k , & 0 x 2 k x, & 2 x 4 array . If _ x 2 f(x) exists, then what is the value of k ?

Step-by-step solution

f(x)= cases 1+ x 2 k , & 0 x 2 k x, & 2 x 4 cases Hence, _ x 2 f(x) exists so we can write that aligned & _ x 2⁻ f(x)= _ x 2⁺ f(x) & _ x 2⁻ (1+ x 2 k )= _ x 2⁺ (k x) & 1+ 2 2 k =2 k & 1+ 1 k =2 k & k+1=2 k^2 & 2 k^2-k-1=0 & 2 k^2-2 k+k-1=0 & 2 k(k-1)+1(k-1)=0 & (2 k+1)(k-1)=0 & k=1 or k=- 1 2 aligned