Convex and concave curves and points of inflectionAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^3-6x^2+5x$.

   (a)

   Find the $x$-coordinate of the point of inflection of $C$.

   [1 mark]

   • A$x=-2$
   • B$x=2$
   • C$x=\frac12$
   • D$x=6$

   (b)

   Find the set of values of $x$ for which $C$ is convex.

   [1 mark]

   • A$x<2$
   • B$x>6$
   • C$x>2$
   • D$x>-2$

   (c)

   Find the equation of the tangent to $C$ at its point of inflection.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x^4-4x^3+6$.

   (a)

   Find $\frac{d^2y}{dx^2}$.

   [1 mark]

   • A$4x^3-12x^2$
   • B$12x^2-12x$
   • C$24x-24$
   • D$12x^2-24x$

   (b)

   Find the set of values of $x$ for which $C$ is concave.

   [1 mark]

   • A$0<x<2$
   • B$x<0$ or $x>2$
   • C$x<2$
   • D$x>2$

   (c)

   Find the coordinates of both points of inflection of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^2+\frac{8}{x}$ for $x\neq0$.

   (a)

   Show that $C$ has exactly one point of inflection and find its coordinates.

   [3 marks]

   (b)

   Find the set of values of $x$ for which $C$ is concave.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^3+ax^2+bx$, where $a$ and $b$ are constants. $C$ has a point of inflection at the point where $x=1$, and the gradient of $C$ at that point is $-5$.

   (a)

   (i) Show that $a=-3$.
   (ii) Find the value of $b$ and the coordinates of the point of inflection.

   [6 marks]

   (b)

   (i) Find the set of values of $x$ for which $C$ is convex and explain what this means for the gradient of $C$.
   (ii) Explain why the gradient of $C$ is least at the point of inflection, and state the least gradient.

   [6 marks]

   Total for question 4: 12 marks

End of questions