Quick questions on Applications of differentiation: tangents, stationary points and optimisation - OCR A-Level Maths A

A point of inflection is where the curve changes the way it bends, that is where $\dfrac{d^2y}{dx^2}$ changes sign. A stationary point of inflection has $\dfrac{dy}{dx} = 0$ and $\dfrac{d^2y}{dx^2} = 0$ with the same sign of the first derivative either side.

In optimisation you write the quantity to be maximised or minimised as a function of one variable (using a constraint to eliminate the other), differentiate, set the derivative to zero, solve, and confirm the nature of the stationary point. Always finish by answering the question that was asked (the optimal value, not just the variable).

When two quantities both depend on time, the chain rule links their rates: $\dfrac{dV}{dt} = \dfrac{dV}{dr}\cdot\dfrac{dr}{dt}$. Identify the rate you know, the rate you want, and the relationship between the quantities.

Find the equation of the tangent to $y = x^2 - 3x$ at $x = 2$. [3 marks]

Find the $x$-coordinate of the minimum of $y = x^2 - 8x + 5$. [2 marks]