Equating 12x^2 + 4 =0, we have x^2 = - \dfrac{1}{3}. We are unable to find a (real) root for this, so conclude that there are no points of inflexion.

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Note: If there is a point of inflexion that is also a stationary point (i.e. f'(x) = 0 also), then it is called a stationary point of inflexion.

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Note: While all points of inflexion have f''(x) = 0, not all points where f''(x) = 0 are points of inflexion. We have to check the curve actually changes from convex to concave or vice versa by seeing what happens on either side of the point.

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When x<2, \dfrac{d^2 y}{dx^2} \textcolor{red}{< 0} and when x>2, \dfrac{d^2 y}{dx^2} \textcolor{blue}{> 0}, so there is a point of inflexion at x=2.

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Question 3: The function f(x) = 2x^4 - 3x^3 has two stationary points. Show that only one of them is a point of inflexion.

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The MME A-A* A level maths practice papers are excellent for those top achieving students to practise for their exams, using authentic exam style questions that are unique to our practice papers. Our content experts have studied A level maths past papers and specifications to develop specific A-A* A level maths exam questions in an authentic exam style. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country.

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