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 +0.6

1.07s Parametric and implicit differentiation
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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 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 H240/01 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.
OCR C4 2005 June Q6
8 marks Standard +0.3
The equation of a curve is \(xy^2 = 2x + 3y\).
  1. Show that \(\frac{dy}{dx} = \frac{2 - y^2}{2xy - 3}\). [5]
  2. Show that the curve has no tangents which are parallel to the \(y\)-axis. [3]
AQA Paper 3 2019 June Q9
15 marks Challenging +1.2
A curve has equation $$x^2y^2 + xy^4 = 12$$
  1. Prove that the curve does not intersect the coordinate axes. [2 marks]
    1. Show that \(\frac{dy}{dx} = -\frac{2xy + y^3}{2x^2 + 4xy^2}\) [5 marks]
    2. Prove that the curve has no stationary points. [4 marks]
    3. In the case when \(x > 0\), find the equation of the tangent to the curve when \(y = 1\) [4 marks]