Çözüldü Cubic Function Derivative - Lines in Analytic Geometry - Inequality with Quadratic Equations

In which interval should m be so that all tangents drawn to the function f(x) = x^3 / 3 + m·x^2 + 6m·x + 1 always make an acute angle with the x-axis in the positive direction?

A) (-6, 10)
B) (-4, 0)
C) (6, 10)
D) (0, 12)
E) (0, 6)

The condition f '(x) > 0 must be met, so to make f '(x) = 1·x^2 + 2m·x + 6m > 0, 1 > 0 and (half) discriminant Δ = m^2 - 6m < 0
m(m - 6) < 0 ⇒ m > 0 and m < 6, then the required interval for m is (0, 6).