5.6 Stationary pointsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The function $f$ is given by $f(x)=2x^3-9x^2+12x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$6x^2-9x+12$
   • B$6x^2-18x+12$
   • C$6x^3-18x^2+12x$
   • D$6x^2-18x$

   (b)

   Find the values of $x$ for which the gradient of the curve is zero.

   [1 mark]

   • A$x=-1$ and $x=-2$
   • B$x=0$ only
   • C$x=1$ and $x=2$
   • D$x=1$ and $x=-2$

   (c)

   Justify that the point where $x=1$ is a local maximum, and write down its coordinates.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The gradient function of a curve $y=f(x)$ is $f'(x)=(x-1)(x+3)$.

   (a)

   Find the values of $x$ at the stationary points of the curve.

   [1 mark]

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

   (b)

   Which statement about the stationary point where $x=-3$ is correct?

   [1 mark]

   • AIt is a local minimum
   • BIt is neither a local maximum nor a local minimum
   • CIt gives the greatest value of $f(x)$ for all $x$
   • DIt is a local maximum

   (c)

   Justify that $f$ has a local minimum at $x=1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company sells $x$ hundred units of a product each week, where $0\le x\le 8$. The weekly profit, $P$ thousand USD, is modelled by $P(x)=-x^3+9x^2-15x+10$.

   (a)

   Find $P'(x)$ and hence find the values of $x$ for which $P'(x)=0$.

   [3 marks]

   (b)

   Justify that the weekly profit has a local maximum when $x=5$, and find this profit in USD.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The temperature in a greenhouse, $T$ °C, $t$ hours after midnight is modelled by $T(t)=-0.02t^3+0.6t^2-3.6t+20$ for $0\le t\le 24$.

   (a)

   (i) Find $T'(t)$.
   (ii) Use your GDC to solve $T'(t)=0$.
   (iii) State which solution gives a local minimum of $T$, justifying your answer using the sign of $T'(t)$.

   [6 marks]

   (b)

   A gardener says: “The lowest temperature in the greenhouse during the 24 hours is at the local minimum, and the highest temperature is at the local maximum.” Use your GDC to evaluate $T$ at the stationary points and at the ends of the domain, and decide whether each part of the gardener’s statement is correct.

   [6 marks]

   Total for question 4: 12 marks

End of questions