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Question
Test whether the following functions are increasing or decreasing : f(x) = x3 – 6x2 + 12x – 16, x ∈ R.
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Solution
f(x) = x3 – 6x2 + 12x – 16
∴ f'(x) = `"d"/"dx" ("x"^3 - "6x"^2 + "12x" - 16)`
= 3x2 – 6 × 2x + 12 × 1 – 0
= 3x2 – 12x + 12
= 3(x2 – 4x + 4)
= 3(x - 2)2 ≥ 0 for all x ∈ R
∴ f'(x) ≥ 0 for all x ∈ R
∴ f is increasing for all x ∈ R.
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