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Question
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Solution
f(x) = 2 – 3x + 3x2 – x3
∴ f′(x) = – 3 + 6x – 3x2
= –3(x2 – 2x + 1)
= –3(x – 1)2
(x – 1)2 is always positive for x ≠ 1 and – 3 < 0.
∴ –3(x – 1)2 is always negative for x ≠ 1.
∴ f′(x) ≤ 0 for all x ∈ R.
Hence, f(x) is a decreasing function for all x ∈ R.
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