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Question
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Options
increasing for all x ∈ R, x ≠ 1
decreasing
neither, increasing nor decreasing
decreasing for all x ∈ R, x ≠ 1
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Solution
increasing for all x ∈ R, x ≠ 1
Explanation:
f(x) = x3 - 3x2 + 3x - 100
Differentiating w.r.t. x, we get
f'(x) = 3x2 - 6x + 3
= 3(x2 - 2x + 1)
= 3(x - 1)2
Note that (x – 1)2 > 0 for all x ∈ R, x ≠ 1.
∴ 3(x - 1)2 > 0 for all x ∈ R, x ≠ 1
∴ f(x) is increasing for all x ∈ R, x ≠ 1.
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