Stationary points and increasing and decreasing functionsEdexcel International 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$x=0$ and $x=3$
   • D$x=2$ only

   (b)

   Determine the nature of the stationary point of $C$ at $x=1$.

   [1 mark]

   • AMinimum, because $\frac{d^2y}{dx^2}=-6$
   • BMaximum, because $\frac{d^2y}{dx^2}=-6<0$
   • CMaximum, because $\frac{dy}{dx}=0$
   • DPoint of inflection, because $\frac{d^2y}{dx^2}\neq0$

   (c)

   Find the $y$-coordinates of the two stationary points of $C$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2x^3-3x^2-12x+5$.

   (a)

   Find the set of values of $x$ for which $\mathrm{f}$ is decreasing.

   [1 mark]

   • A$x<-1$ or $x>2$
   • B$-2<x<1$
   • C$-1<x<2$
   • D$x<2$

   (b)

   Find the value of $\mathrm{f}(x)$ at its local minimum point.

   [1 mark]

   • A$12$
   • B$-2$
   • C$5$
   • D$-15$

   (c)

   Hence find the range of values of $k$ for which the equation $\mathrm{f}(x)=k$ has three distinct real roots.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x+\frac{4}{x^2}$, for $x>0$.

   (a)

   Find $\frac{dy}{dx}$ and hence find the coordinates of the stationary point of $C$.

   [3 marks]

   (b)

   Find $\frac{d^2y}{dx^2}$ and hence show that the stationary point is a minimum.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x^3-3x^2-9x+k$, where $k$ is a constant. $C$ has a stationary point $B$ with $x>0$, at which $y=-20$.

   (a)

   Find the $x$-coordinates of the two stationary points of $C$ and determine the nature of each.

   [6 marks]

   (b)

   (i) Show that $k=7$.
   (ii) Find the coordinates of the other stationary point of $C$.
   (iii) Find the set of values of $c$ for which the equation $x^3-3x^2-9x+7=c$ has exactly one real root.

   [6 marks]

   Total for question 4: 12 marks

End of questions