5.2 Increasing and decreasing functionsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A function $f$ has derivative $f'(x)=x^2+x-6$.

   (a)

   Which statement about $f$ at $x=1$ is correct?

   [1 mark]

   • AIncreasing, with gradient $4$
   • BIncreasing, with gradient $-4$
   • CDecreasing, with gradient $4$
   • DDecreasing, with gradient $-4$

   (b)

   Which of the following is an interval on which $f$ is decreasing?

   [1 mark]

   • A$-3<x<2$
   • B$x<-3$
   • C$x>2$
   • D$x<2$

   (c)

   Find the set of values of $x$ for which $f$ is increasing.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A continuous function $g$ is defined for $-4\le x\le 6$. Its derivative satisfies $g'(x)>0$ for $-4<x<-1$, $g'(x)=0$ at $x=-1$ and at $x=3$, $g'(x)<0$ for $-1<x<3$, and $g'(x)>0$ for $3<x<6$.

   (a)

   On which interval is $g$ decreasing?

   [1 mark]

   • A$-4<x<-1$
   • B$-1<x<3$
   • C$3<x<6$
   • D$-4<x<3$

   (b)

   Which of the following statements about the gradient of $g$ is correct?

   [1 mark]

   • A$g'(-2)<0$
   • B$g'(5)<0$
   • C$g'(0)<0$
   • D$g'(3)>0$

   (c)

   Given that $g(-1)=7$, explain why $g(2)<7$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The height, $h$ metres, of a drone $t$ seconds after take-off is modelled by $h(t)=t^3-12t^2+36t+5$, for $0\le t\le 8$.

   (a)

   Find $h'(t)$. Hence use your GDC to find the values of $t$ for which $h'(t)=0$.

   [3 marks]

   (b)

   Hence find the intervals of time during which the drone is rising and the interval during which it is descending.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The daily readership, $R$ thousand, of a local newspaper $t$ years after it was launched is modelled by $R(t)=-t^3+15t^2-63t+200$, for $0\le t\le 10$.

   (a)

   (i) Find $R'(t)$.
   (ii) Use your GDC to find the values of $t$ for which $R'(t)=0$.
   (iii) Hence find the values of $t$ for which the readership is increasing, justifying your answer.

   [6 marks]

   (b)

   The owner claims that the readership never falls below 100 thousand during the 10 years. Use the intervals on which $R$ is increasing and decreasing to decide whether the claim is correct.

   [6 marks]

   Total for question 4: 12 marks

End of questions