Show no stationary points exist

A question is this type if and only if it asks to prove that a curve has no stationary points by showing dy/dx ≠ 0 everywhere.

7 questions · Standard +0.5

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CAIE P2 2020 November Q5
9 marks Standard +0.3
5 The equation of a curve is \(2 \mathrm { e } ^ { 2 x } y - y ^ { 3 } + 4 = 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 \mathrm { e } ^ { 2 x } y } { 3 y ^ { 2 } - 2 \mathrm { e } ^ { 2 x } }\).
  2. The curve passes through the point \(( 0,2 )\). Find the equation of the tangent to the curve at this point, giving your answer in the form \(a x + b y + c = 0\).
  3. Show that the curve has no stationary points.
CAIE P2 2014 June Q7
10 marks Standard +0.3
7 The equation of a curve is $$2 x ^ { 2 } + 3 x y + y ^ { 2 } = 3$$
  1. Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  2. Show that the curve has no stationary points.
CAIE P3 2024 March Q6
7 marks Standard +0.3
6 The equation of a curve is \(2 y ^ { 2 } + 3 x y + x = x ^ { 2 }\).
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 2 \mathrm { x } - 3 \mathrm { y } - 1 } { 4 \mathrm { y } + 3 \mathrm { x } }\).
  2. Hence show that the curve does not have a tangent that is parallel to the \(x\)-axis.
OCR C4 2005 June Q6
8 marks Standard +0.3
6 The equation of a curve is \(x y ^ { 2 } = 2 x + 3 y\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - y ^ { 2 } } { 2 x y - 3 }\).
  2. Show that the curve has no tangents which are parallel to the \(y\)-axis.
OCR H240/01 2022 June Q7
8 marks Standard +0.8
7 A curve has equation \(2 x ^ { 3 } + 6 x y - 3 y ^ { 2 } = 2\).
Show that there are no points on this curve where the tangent is parallel to \(y = x\).
OCR Pure 1 2018 March Q10
10 marks Challenging +1.2
10 In this question you must show detailed reasoning.
Show that the curve with equation \(x ^ { 2 } - 4 x y + 8 y ^ { 3 } - 4 = 0\) has exactly one stationary point.
AQA Paper 3 2019 June Q9
15 marks Standard +0.3
9 A curve has equation $$x ^ { 2 } y ^ { 2 } + x y ^ { 4 } = 12$$ 9
  1. Prove that the curve does not intersect the coordinate axes.
    9
    1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 x y + y ^ { 3 } } { 2 x ^ { 2 } + 4 x y ^ { 2 } }\) 9
  3. (ii) Prove that the curve has no stationary points.
    9
  4. (iii) In the case when \(x > 0\), find the equation of the tangent to the curve when \(y = 1\)