Find the anti derivative $F$ of $f$ defined by $f(x) = 4x^3 - 6$, where $F(0) = 3$

$x^4 - 6x + 3$

The correct answer is Option (1) → $x^4 - 6x + 3$

One anti derivative of $f(x)$ is $x^4 - 6x$ since

$\frac{d}{dx} (x^4 - 6x) = 4x^3 - 6$

Therefore, the anti derivative $F$ is given by

$F(x) = x^4 - 6x + C, \text{ where } C \text{ is constant.}$

Given that $F(0) = 3$, which gives,

$3 = 0 - 6 \times 0 + C \quad \text{or} \quad C = 3$

Hence, the required anti derivative is the unique function $F$ defined by

$F(x) = x^4 - 6x + 3.$