The curve $y=2x^3+a{x}^2+bx-30$ has no stationary points. Show that $a^2<6b$.

A curve has equation $y=1+2x+\frac{k^2}{2x-3}$, where $k$ is a positive constant. Find, in terms of $k$, the values of $x$ for which the curve has stationary points and determine the nature of each stationary point.

Find the coordinates of the stationary points on the curve $y=x^4-4x^3+4x^2+1$ and determine the nature of each of these points. Sketch the graph of the curve.

The curve $y=x^3+a{x}^2+b$ has a stationary point at $(4,-27)$. Find the values of $a$ and $b$.

The curve $y=x^3+a{x}^2+b$ has a stationary point at $(4,-27)$. Determine the nature of the stationary point $(4,-27)$.

The curve $y=x^3+a{x}^2+b$ has a stationary point at $(4,-27)$. Find the coordinates of the other stationary point on the curve and determine the nature of this stationary point.

The curve $y=x^3+a{x}^2+b$ has a stationary point at $(4,-27)$. Find the coordinates of the point on the curve where the gradient is minimum and state the value of the minimum gradient.

The curve $y=ax+\frac{b}{x^2}$ has a stationary point at $(2,12)$. Find the values of $a$ and $b$.

The curve $y=ax+\frac{b}{x^2}$ has a stationary point at $(2,12)$. Determine the nature of the stationary point $(2,12)$.