Determine constant from stationary point condition

A question is this type if and only if it gives a curve with an unknown constant and states that a particular point is a stationary point (or gives the gradient there), requiring the constant to be found before further analysis.

17 questions · Standard +0.0

1.07n Stationary points: find maxima, minima using derivatives
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Edexcel P2 2020 January Q10
10 marks Standard +0.3
10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),
  1. show that \(k = - \frac { 3 } { 2 }\)
  2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
  3. Using algebra, find the \(x\) coordinate of this second stationary point. \includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}
OCR C1 2008 June Q8
10 marks Moderate -0.3
8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
  3. Determine whether this stationary point is a minimum or maximum point.
  4. Find the \(x\)-coordinate of the other stationary point.
OCR C1 2009 January Q9
7 marks Moderate -0.3
9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.
OCR C1 2015 June Q9
10 marks Moderate -0.3
9 The curve \(y = 2 x ^ { 3 } - a x ^ { 2 } + 8 x + 2\) passes through the point \(B\) where \(x = 4\).
  1. Given that \(B\) is a stationary point of the curve, find the value of the constant \(a\).
  2. Determine whether the stationary point \(B\) is a maximum point or a minimum point.
  3. Find the \(x\)-coordinate of the other stationary point of the curve.
OCR C1 2016 June Q11
8 marks Standard +0.3
11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .
Edexcel Paper 2 2023 June Q5
5 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)
The curve
  • passes through the point \(P ( 3 , - 10 )\)
  • has a turning point at \(P\)
Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } - 9 x ^ { 2 } + 5 x + k$$ where \(k\) is a constant,
  1. show that \(k = 12\)
  2. Hence find the coordinates of the point where \(C\) crosses the \(y\)-axis.
OCR MEI Paper 1 2024 June Q13
8 marks Moderate -0.8
13 The curve with equation \(\mathrm { y } = \mathrm { px } + \frac { 8 } { \mathrm { x } ^ { 2 } } + \mathrm { q }\), where \(p\) and \(q\) are constants, has a stationary point at \(( 2,7 )\).
  1. Determine the values of \(p\) and \(q\).
  2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence determine the nature of the stationary point at (2, 7).
OCR H240/03 2018 September Q2
6 marks Standard +0.3
2 A curve has equation \(y = a x ^ { 4 } + b x ^ { 3 } - 2 x + 3\).
  1. Given that the curve has a stationary point where \(x = 2\), show that \(16 a + 6 b = 1\).
  2. Given also that this stationary point is a point of inflection, determine the values of \(a\) and \(b\).
OCR AS Pure 2017 Specimen Q4
7 marks Moderate -0.8
4 The curve \(y = 2 x ^ { 3 } + 3 x ^ { 2 } - k x + 4\) has a stationary point where \(x = 2\).
  1. Determine the value of the constant \(k\).
  2. Determine whether this stationary point is a maximum or a minimum point.
OCR H240/03 Q6
7 marks Standard +0.3
6 A curve has equation \(y = x ^ { 2 } + k x - 4 x ^ { - 1 }\) where \(k\) is a constant. Given that the curve has a minimum point when \(x = - 2\)
  • find the value of \(k\)
  • show that the curve has a point of inflection which is not a stationary point.
CAIE P1 2019 June Q8
8 marks Standard +0.3
A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]
CAIE P1 2011 November Q8
10 marks Moderate -0.3
A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.
  1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
  2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
  3. Find \(\mathrm{f}(x)\). [4]
OCR C1 Q7
9 marks Moderate -0.3
A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),
  1. find the values of \(a\) and \(b\), [6]
  2. find the coordinates of the other stationary point of the curve. [3]
AQA AS Paper 2 2020 June Q10
8 marks Standard +0.3
A curve has gradient function $$\frac{dy}{dx} = 3x^2 - 12x + c$$ The curve has a turning point at \((-1, 1)\)
  1. Find the coordinates of the other turning point of the curve. Fully justify your answer. [6 marks]
  2. Find the set of values of \(x\) for which \(y\) is increasing. [2 marks]
SPS SPS SM Pure 2021 June Q15
9 marks Standard +0.8
A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)
  1. find \(g(x)\), [7]
  2. prove that the stationary point at \((2, 9)\) is a maximum. [2]
SPS SPS FM 2020 September Q8
8 marks Challenging +1.2
\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.
  1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
  2. find, showing your working, the integer value of \(k\). [5]
OCR H240/03 2017 Specimen Q6
7 marks Standard +0.3
A curve has equation \(y = x^2 + kx - 4x^{-1}\) where \(k\) is a constant. Given that the curve has a minimum point when \(x = -2\)
  • find the value of \(k\)
  • show that the curve has a point of inflection which is not a stationary point. [7]