Maharashtra State BoardHSC Arts 12th Board Exam
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Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing - Mathematics and Statistics

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Sum

Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing

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Solution

f(x) = 2 – 3x + 3x2 – x

∴ 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.

Concept: Increasing and Decreasing Functions
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