Answer:
4 < k < 12
Step-by-step explanation:
The given function is y = 2·x² + k·x + 2·k - 6
Where the curve lies above the x-axis for all values of x, the function does not have a root and the discriminant, Δ, is negative, to give complex roots
∴ The discriminant Δ = k² - 4·2·(2·k - 6) < 0
k² - 4·2·(2·k - 6) = k² - 16·k + 48 < 0
(k - 12)·(k - 4) < 0
∴ k < 12 or k > 4 given that by substitution, we have;
When k = 3 < 4
k² - 16·k + 48 = 3² - 16×3 + 48 = 9 > 0
Also the product of two negative numbers is larger than 0
Therefore, the set of values of k for which the curve lies above the x-axis for all values of x is 4 < k < 12
