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