Q.11 Test whether the function f(x) = x³ + 6x² + 12x - 7 is increasing or decreasing for all x in R .

Answer: f is increasing for all x in R . f'(x) = 3x² + 12x + 12 = 3(x² + 4x + 4) = 3(x + 2)² . Since (x + 2)² ≥ 0 for all real x , f'(x) ≥ 0 for all x in R (it is zero only at the single point x = -2 ). Hence f is an increasing function for all x in R .

Test whether the function \(f(x) = x^3 + 6x^2 + 12x - 7\) is increasing or decreasing for all \(x \in R\).

Q.11 Test whether the function \(f(x) = x^3 + 6x^2 + 12x - 7\) is increasing or decreasing for all \(x \in R\).

Doubtly Asked 28 Oct 2026 Board exam · March 2026 hsc 12th Save WhatsApp Report

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Answer: \(f\) is increasing for all \(x \in R\).

\(f'(x) = 3x^2 + 12x + 12 = 3(x^2 + 4x + 4) = 3(x + 2)^2\).

Since \((x + 2)^2 \ge 0\) for all real \(x\), \(f'(x) \ge 0\) for all \(x \in R\) (it is zero only at the single point \(x = -2\)). Hence \(f\) is an increasing function for all \(x \in R\).

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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Test whether the function \(f(x) = x^3 + 6x^2 + 12x - 7\) is increasing or decreasing for all \(x \in R\).

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