Find the equations of the tangent and normal to the given curve $y=x^{2}$ at the point $(0,0)$.

At which points on the curve $y = x^3 - 3x^2 - 9x + 7$ is the tangent parallel to the $x$-axis?

If the curves $y = x^3 - 3x^2 - 8x - 4$ and $y = 3x^2 + 7x + 4$ touch each other at a point $P$,then the equation of the common tangent at $P$ is:

Find the slope of the tangents to the curve $y = (x + 1)(x - 3)$ at the points where it meets the $x$-axis.

The curve $y=ax^3+bx^2+cx+5$ touches the $X$-axis at $P(-2,0)$ and cuts the $Y$-axis at a point $Q$,where its gradient is $3$. Then:

Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f(x) = x^3 - x^2 - 2x$ at $(x, y)$ is parallel to the line segment joining the points $(1, f(1))$ and $(-1, f(-1))$. Then $S$ is equal to: