Q.3 (A)(ii) If f'(x) = 4x³ - 3x² + 2x + k,\ f(0) = 1 and f(1) = 4, find f(x).

Answer: f(x) = x⁴ - x³ + x² + 2x + 1 f(x) = ∫(4x³ - 3x² + 2x + k) dx = x⁴ - x³ + x² + kx + c. f(0) = 1 ⇒ c = 1. f(1) = 4 ⇒ 1 - 1 + 1 + k + 1 = 4 ⇒ k = 2. Hence f(x) = x⁴ - x³ + x² + 2x + 1.

Chapter: Integration· Mathematics & Statistics (Commerce) Part 1

If \(f'(x) = 4x^3 - 3x^2 + 2x + k,\ f(0) = 1\) and \(f(1) = 4\), find \(f(x)\).

Q.3 (A)(ii) If \(f'(x) = 4x^3 - 3x^2 + 2x + k,\ f(0) = 1\) and \(f(1) = 4\), find \(f(x)\).

Doubtly Asked 29 Oct 2026 Board exam · March 2024 hsc 12th Save WhatsApp Report

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Answer: \(f(x) = x^4 - x^3 + x^2 + 2x + 1\)

\(f(x) = \int(4x^3 - 3x^2 + 2x + k)\,dx = x^4 - x^3 + x^2 + kx + c\).

\(f(0) = 1 \Rightarrow c = 1\).

\(f(1) = 4 \Rightarrow 1 - 1 + 1 + k + 1 = 4 \Rightarrow k = 2\).

Hence \(f(x) = x^4 - x^3 + x^2 + 2x + 1\).

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report

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If \(f'(x) = 4x^3 - 3x^2 + 2x + k,\ f(0) = 1\) and \(f(1) = 4\), find \(f(x)\).

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