If F is an anti-derivative of \(f(x) = 6x^2+4x-5\) such that \(F(1) = 2\), then \(F(x)\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2x^3+2x^2-5x+3\)
An anti-derivative of f is any function whose derivative is f. Integrating term by term with \(\int x^n\,dx = \frac{x^{n+1}}{n+1}\):
\(F(x) = 2x^3+2x^2-5x+C\)
The condition \(F(1) = 2\) fixes the constant:
\(2+2-5+C = 2\)
\(C = 3\)
So \(F(x) = 2x^3+2x^2-5x+3\).
Why the other options are incorrect
Option (1) \(2x^3+2x^2-5x+2\) — Incorrect. It takes the constant equal to 2, which would give \(F(0) = 2\), not \(F(1) = 2\); here \(F(1) = 1\).
Option (2) \(2x^3+2x^2-5x-1\) — Incorrect. Here \(F(1) = 2+2-5-1 = -2\), not 2.
Option (4) \(12x+4\) — Incorrect. This is the derivative of f, not an anti-derivative.