Mathematical Methods · Unit 3 · Further applications of differentiation · The second derivative and applications of differentiation
AI-marked practice questions tied to QCAA mark schemes for this exact LO. Free to start.
Start free practiceFull questions, answers and worked solutions unlock when you start a free practice session.
Consider the function $f(x) = x^3 - 3x^2 - 9x + 5$ for $x \in \mathbb{R}$. (a) Find the coordinates of the stationary points of $f(x)$. [1 mark] (b) Determine the nature (maximum, minimum, or point of inflection) of each stationary point using the second derivative. [1 mark] (c) State the coordinates of the point of inflection and verify that $f''(x) = 0$ at this point. [1 mark]
Consider the function $f(x) = x^3 - 6x^2 + 9x + 2$ defined on the domain $x \in \mathbb{R}$. The first derivative is $f'(x) = 3x^2 - 12x + 9$ and the second derivative is $f''(x) = 6x - 12$. Which of the following statements correctly identifies both a stationary point and a point of inflection of $f(x)$?
Consider the function $f(x) = x^3 - 6x^2 + 9x + 2$ for $x \in \mathbb{R}$. (a) Find the coordinates of any stationary points. (1 mark) (b) Determine whether each stationary point is a maximum, minimum, or point of inflection by using the second derivative test. (1 mark) (c) State the coordinates of any points of inflection and explain your reasoning. (1 mark)
Consider the function $f(x) = x^3 - 3x^2 - 9x + 5$ defined on the domain $-3 \leq x \leq 5$. Which graph correctly shows the location of the stationary points and the point of inflection?
Consider the function $f(x) = x^3 - 6x^2 + 9x + 2$ for $x \in \mathbb{R}$. (a) Find the coordinates of all stationary points. (1 mark) (b) Determine which stationary points are local maxima, local minima, or horizontal points of inflection by evaluating the second derivative. (1 mark) (c) State the coordinates of any points of inflection and describe the behaviour of the function near each point. (1 mark)