Tangents, Normals & Curve Features — practice questions

Find the equation of the tangent to $y = x^2 - 4x + 1$ at the point where $x = 3$.

Find the equation of the normal to $y = x\ln x$ at the point where $x = e$, giving your answer in the form $y = mx + c$.

Two tangents to the curve $y = x^2$ are drawn, one at $x = a$ and one at $x = b$, where $a \neq b$. Show that they intersect at the point $\left(\dfrac{a+b}{2},\; ab\right)$, and hence find the condition on $a$ and $b$ for the two tangents to be perpendicular.

Find the coordinates of the stationary points of $y = x^3 - 12x + 4$.

Find the stationary points of $y = x^4 - 4x^3$ and determine the nature of each, justifying your conclusions.

The curve $y = x^3 + ax^2 + bx$ has a stationary point at $(2, -8)$. Find $a$ and $b$, determine the nature of that stationary point, and find the coordinates of the other stationary point.

Find the coordinates of the point of inflection of $y = x^3 - 6x^2 + 2x - 1$.

Show that $y = x^4 + 2x^2$ has no points of inflection, despite having a stationary point where $f''$ is not obviously non-zero.

Prove that for any cubic $y = ax^3 + bx^2 + cx + d$ with $a \neq 0$, the point of inflection lies exactly midway between the two turning points whenever they exist.