If f:[0,2) [R is defined by f(x)= array cl 1+ 2 x k & for 0 x 0 , and f is such that _ x 1⁻ f(x)= _ x 1⁺ f(x) , then

Step-by-step solution

Given function f(x)= array ll 1+ 2 x k , & 0 x 0 . Now, _ x 1⁻ f(x)= _ x 1⁻ (1+ 2 x k )=1+ 2 k and _ x 1⁺ f(x)= _ x 1⁺ (k x)=k Now, given that array rlrl & & _ x 1⁻ f(x) & = _ x 1⁺ f(x) 1+ 2 k & =k & & k+2 k & =k & & k^2-k-2 & =0 k^2-2 k+k-2 & =0 array array rlrl & k(k-2)+1(k-2) & =0 & (k-2)(k+1) & =0 & So, & k & =2, k=-1 & but & k & >0 & So, & k & =2 & k^2 & =4 array