Second derivative test justification

A question is this type if and only if it explicitly requires finding d²y/dx² and using its sign at a stationary point to justify the nature of that point as part of the answer.

24 questions · Moderate -0.5

1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives
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Edexcel C12 2016 January Q10
10 marks Moderate -0.3
10. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the coordinates of the stationary point on \(C\).
  3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.
Edexcel C12 2017 June Q4
12 marks Moderate -0.3
4. The curve \(C\) has equation \(y = 4 x \sqrt { x } + \frac { 48 } { \sqrt { x } } - \sqrt { 8 } , x > 0\)
  1. Find, simplifying each term,
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Use part (a) to find the exact coordinates of the stationary point of \(C\).
  3. Determine whether the stationary point of \(C\) is a maximum or minimum, giving a reason for your answer.
Edexcel P2 2021 January Q2
7 marks Moderate -0.8
2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$
  1. Using calculus, find the \(x\) coordinates of the stationary points of the curve.
  2. Justify, by further calculus, the nature of all of the stationary points of the curve.
Edexcel C2 2010 January Q9
10 marks Moderate -0.8
9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)
  1. Use calculus to find the coordinates of the turning point on \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. State the nature of the turning point.
Edexcel C2 2013 January Q8
9 marks Moderate -0.3
8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)
  1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
  2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).
Edexcel C2 2014 January Q3
11 marks Moderate -0.8
3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$
  1. Find
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Use calculus to find the coordinates of the stationary point of \(C\).
  3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}
Edexcel C2 2006 June Q10
14 marks Moderate -0.3
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-14_636_956_285_513}
\end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).
  1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
    The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
  3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
  4. Hence calculate the exact area of \(R\).
Edexcel C2 2013 June Q9
9 marks Moderate -0.3
  1. The curve with equation
$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus
  1. to find the coordinates of \(P\),
  2. to determine the nature of the stationary point \(P\).
OCR C1 Q7
9 marks Moderate -0.8
7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
  3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  4. Determine whether each stationary point is a maximum or a minimum point.
OCR C1 Q8
9 marks Moderate -0.3
8. $$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$
  1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
  2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
  3. Determine whether the turning point is a maximum or minimum point.
OCR MEI C2 2005 January Q10
11 marks Moderate -0.8
10 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 2005 June Q4
5 marks Moderate -0.8
4 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.
OCR MEI C2 Q10
12 marks Moderate -0.3
10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-5_522_1019_403_394} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence calculate the coordinates of the stationary point on the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
  4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
  5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.
OCR MEI C2 Q2
13 marks Standard +0.3
2 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).
  1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
  2. Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 . Find the coordinates of the other point, P , on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P . Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.
OCR MEI C2 Q3
5 marks Moderate -0.8
3 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.
OCR MEI C2 Q2
11 marks Moderate -0.8
2 A curve has equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 12\).
  1. Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
  2. Find, in the form \(y = m x + c\), the equation of the normal to the curve at the point \(( 2 , - 4 )\).
OCR MEI C2 Q3
13 marks Standard +0.3
3 A cubic curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 1\).
  1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.
  2. Show that the tangent to the curve at the point where \(x = - 1\) has gradient 9 . Find the coordinates of the other point, P , on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P . Show that the area of the triangle bounded by the normal at P and the \(x\) - and \(y\)-axes is 8 square units.
OCR MEI C3 Q4
7 marks Moderate -0.3
4
  1. Show that \(y = a x e ^ { - x }\) for \(a > 0\) has only one stationary point for all values of \(x\). Determine whether this stationary value is a maximum or minimum point.
  2. Sketch the curve.
OCR C1 2011 June Q8
7 marks Moderate -0.3
8
  1. Find the coordinates of the stationary point on the curve \(y = 3 x ^ { 2 } - \frac { 6 } { x } - 2\).
  2. Determine whether the stationary point is a maximum point or a minimum point.
OCR MEI AS Paper 2 2020 November Q9
8 marks Moderate -0.8
9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).
  1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Find the coordinates of the turning point.
  3. Determine the nature of the turning point.
AQA C1 2010 January Q3
12 marks Moderate -0.8
3 The depth of water, \(y\) metres, in a tank after time \(t\) hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$
  1. Find:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
  2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether it is a maximum value or a minimum value.
    1. Find the rate of change of the depth of water, in metres per hour, when \(t = 1\).
    2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when \(t = 1\).
AQA C1 2011 January Q1
10 marks Moderate -0.8
1 The curve with equation \(y = 13 + 18 x + 3 x ^ { 2 } - 4 x ^ { 3 }\) passes through the point \(P\) where \(x = - 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that the point \(P\) is a stationary point of the curve and find the other value of \(x\) where the curve has a stationary point.
    1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
    2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
      (l mark)
AQA C2 2012 June Q6
10 marks Moderate -0.8
6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.
    1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
    2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
    3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
  2. Find the equation of the curve.
Pre-U Pre-U 9794/1 2012 Specimen Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinates of the stationary points of the curve.
  3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.