Stationary points and increasing and decreasing functionsAQA A-Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=x^3-6x^2+9x+2$.

   (a)

   Find the $x$-coordinates of the stationary points of $C$.

   [1 mark]

   • A$x=1$ and $x=3$
   • B$x=-1$ and $x=-3$
   • C$x=2$
   • D$x=1$ and $x=-3$

   (b)

   Find the coordinates of the maximum point of $C$.

   [1 mark]

   • A$(3,2)$
   • B$(1,6)$
   • C$(1,0)$
   • D$(2,4)$

   (c)

   Find the set of values of $x$ for which $y$ is decreasing.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=x+\frac{16}{x}$ for $x>0$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

   • A$1+\frac{16}{x^2}$
   • B$-\frac{16}{x^2}$
   • C$1-\frac{16}{x^2}$
   • D$1-\frac{32}{x^3}$

   (b)

   Find the value of $\frac{d^2y}{dx^2}$ when $x=4$.

   [1 mark]

   • A$2$
   • B$-\frac12$
   • C$8$
   • D$\frac12$

   (c)

   Find the minimum value of $y$, justifying that it is a minimum.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An open-topped box has a square base of side $x$ cm and height $h$ cm. It is made from $300$ cm$^2$ of card, so the base and four sides use all of the card. The volume of the box is $V$ cm$^3$.

   (a)

   Show that $V=75x-\frac{x^3}{4}$.

   [3 marks]

   (b)

   Find the maximum volume of the box, justifying that it is a maximum.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^3+ax^2-24x+7$, where $a$ is a constant. $C$ has a stationary point at the point where $x=4$.

   (a)

   (i) Show that $a=-3$.
   (ii) Find the coordinates of the other stationary point of $C$.

   [6 marks]

   (b)

   Use the second derivative to determine the nature of each stationary point of $C$, and hence find the set of values of $x$ for which $y$ is increasing.

   [6 marks]

   Total for question 4: 12 marks

End of questions