Consider the function f : 1 2 , 1 → R defined by f x = 4 2 x 3 - 3 2 x - 1 . Consider the statements (I) The curve y = f x intersects the x -axis exactly at one point (II) The curve y = f x intersects the x -axis at x = cos π 12 Then

Step-by-step solution

Given: f x = 4 2 x 3 - 3 2 x - 1 ⇒ f ' x = 12 2 x 2 - 3 2 ⇒ f ' x = 3 2 4 x 2 - 1 ⇒ f ' x = 3 2 2 x - 1 2 x + 1 So, the function is increasing in x ∈ 1 2 , 1 . Thus, the function will intersect the x - axis only at one point. Hence, statement I is correct. Now, putting x = π 12 in the equation cos 3 x = 4 cos 3 π 12 - 3 cos π 12 . ⇒ cos π 4 = 4 cos 3 π 12 - 3 cos π 12 ⇒ 4 2 cos 3 π 12 - 3 2 cos π 12 - 1 = 0 Now, comparing with f x = 4 2 x 3 - 3 2 x - 1 we get. cos π 12 is a solution of f x . So, statement- II is al