Let f(x)= cases & (x-a)^2+b, & x k [2 x], & 0 < x < k ⁻¹ ( 1-x^2 1+x^2 ), & x 0 cases If f(x) is non-derivable at exactly 6 points, then the value of a b k is: [Note: [k] denotes greatest integer less than or equal to k and k N .]

Step-by-step solution

f(x) is non differentiable at x=0 and [2 x] is discontinuous at x= 1 2 , 1, 3 2 , 2, 5 2 , 3 _ 5 points f(x) should be differentiable and continuous at x=2 , so k=3 aligned & f^ (3⁺ )=f^ (3⁻ ) 2(3-a)=0, a=3 & f (3⁺ )=f (3⁻ ) (3-a)^2+b=5, b=5 & a b k=3 5 3=45 aligned