If m is the minimum value of k for which the function f x = x k x - x 2 is increasing in the interval [ 0, 3 ] and M is the maximum value of f in [ 0, 3 ] when k = m , then the ordered pair ( m , M ) is equal to:

Step-by-step solution

f x = x k x - x 2 ⇒   f ' x = 3 k x - 4 x 2 2 k x - x 2 As per the given condition f ' x ≥ 0 for x ∈ 0,   3 ⇒ 3 k x - 4 x 2 ≥ 0 for x ∈ 0,   3 ⇒ 3 k - 4 x   ≥ 0 for x ∈ 0,   3 ⇒ k ≥ 4 x 3 for x ∈ 0,   3 ⇒ k ≥ 4 . So minimum value of k is m = 4 . Now f x = x 4 x - x 2 Since given function in increasing hence maximum value will occur at x = 3 ⇒ f ( 3 ) = 3 4 × 3 - 3 2 = 3 3 ,   M