Maharashtra State BoardHSC Arts 12th Board Exam
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Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R - Mathematics and Statistics

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Sum

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.

Concept: Increasing and Decreasing Functions
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