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This is a Differentiation question.

iii)

y = f (x) + kx = 2x^3 + x^2 – 8x + kx – 4

For turning point 

dy/dx = 6x^2 + 2x – 8 +k = 0

Using discriminant method,

(2)^2 – 4 (6) (k – 8) >= 0 (2 turning point)

4 – 24k + 48 >= 0

52 >= 24k

k <= 52/24 = 13/6

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