If 6 x 2 - 5 x - 3 x 2 - 2 x + 6 ≤ 4 , then the least and highest values of 4 x 2 are

Step-by-step solution

We have, 6 x 2 - 5 x - 3 x 2 - 2 x + 6 ≤ 4 ⇒ 6 x 2 - 5 x - 3 - 4 x 2 + 8 x - 24 x 2 - 2 x + 6 ≤ 0 ⇒ 2 x 2 + 3 x - 27 x 2 - 2 x + 6 ≤ 0 ⇒ 2 x + 9 x - 3 x - 1 2 + 5 ≤ 0 Now, x - 1 2 + 5 > 0 , ∀   x ∈ R . So, 2 x 2 + 3 x - 27 ≤ 0 ⇒ 2 x + 9 x - 3 ≤ 0 ⇒ - 9 2 ≤ x ≤ 3 Now, at x = - 9 2 , 4 x 2 = 4 × - 9 2 2 = 81 And, at x = 3 , 4 x 2 = 36 Therefore, greatest value of 4 x 2 is 81 . And, least value of 4 x 2 is 0 occurs