Question

The slope at any point of a curve $y=f\left(x\right)$ is given by $\frac{dy}{dx}=3x^2$ and it passes through $(-1,1)$. The equation of the curve is

• A. $y=x^3+2$
• B. $y=-x^3-2$
• C. $y=3x^3+4$
• D. $y=-x^3+2$

Answer 1

The correct option is A $y=x^3+2$

Given, $\frac{dy}{dx}=3x^2$
$\Rightarrow dy=3x^{2}dx$
On integrating, we get
$y=\frac{3x^3}{3}+c$
$\Rightarrow y=x^3+c$
It passes through $(-1,1)$.
$\therefore 1={(-1)}^3+c$
$\Rightarrow c=2$
$\therefore y=x^3+2$