Question

Test whether the function f(x) = x³ + 6x² + 12x − 5 is increasing or decreasing for all x ∈ R

Answer 1

f(x) = x³ + 6x² + 12x − 5

∴ f′(x) = 3x² + 12x + 12

= 3(x² + 4x + 4)

= 3(x + 2)²

3(x + 2)² is always positive for x ≠ – 2

∴ f′(x) ≥ 0 for all x ∈ R

Hence, f(x) is an increasing function for all x ∈ R.