Paragraph: Let f(x)= 4 + ⁻¹ ( x 1+x^2 )- ⁻¹ x and a_i (a_i Question: If P(x)=x^2-4 k x+3 k^2 is negative for all values of x lying in the interval (a₁, a₂ ) then set of real values of k is:

Step-by-step solution

aligned & f(x)= 4 + ⁻¹ ( x 1+x^2 )- ⁻¹ x & f(x)= 4 + ⁻¹ x- ⁻¹ x & = 3 4 -2 ⁻¹ x x>0 & sgn (f(x))=1 f(x)>0 & 3 4 -2 ⁻¹ x>0 ⁻¹ x < 3 8 x < 2 +1 aligned Possible positive integral values of x are 1,2 a₁=1 and a₂=2 (ii) aligned & P(x)=x^2-4 k x+3 k^2 & P(x) < 0 x (a₁, a₂ ) & P(1) 0 and P(2) 0 & 1-4 k+3 k^2 0 and 4-8 k+3 k^2 0 aligned (3 k-1)(k-1) 0 (3 k-2)(k-2) 0 k [ 1 3 , 1 ] ... (1) k [ 2 3 , 2 ] Intersection of eqns. (1) and (2) is k [ 2 3 , 2 ]