Find stationary points and nature

Questions requiring finding coordinates of stationary points by solving dy/dx = 0 and determining their nature using the second derivative test or sign change of first derivative.

30 questions · Moderate -0.1

1.07n Stationary points: find maxima, minima using derivatives
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Edexcel PURE 2024 October Q3
Standard +0.3
  1. In this question you must show all stages of your working.
$$f ( x ) = \frac { ( x + 5 ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$
  1. Find \(\int f ( x ) d x\)
    1. Show that when \(\mathrm { f } ^ { \prime } ( x ) = 0\) $$3 x ^ { 2 } + 10 x - 25 = 0$$
    2. Hence state the value of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) = 0$$
OCR C1 Q8
11 marks Moderate -0.3
$$\text{f}(x) \equiv \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
  2. Show that $$\text{f}'(x) = \frac{3x^2 - 8x - 16}{4x^{\frac{3}{2}}}.$$ [5]
  3. Find the coordinates of the stationary point of the curve \(y = \text{f}(x)\). [3]
OCR MEI C3 Q6
Standard +0.8
A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve.
OCR MEI C3 2013 January Q1
6 marks Moderate -0.3
  1. Given that \(y = e^{-x} \sin 2x\), find \(\frac{dy}{dx}\). [3]
  2. Hence show that the curve \(y = e^{-x} \sin 2x\) has a stationary point when \(x = \frac{1}{2} \arctan 2\). [3]
SPS SPS SM Pure 2023 October Q2
11 marks Standard +0.3
  1. The curve \(C\) has equation $$y = \frac{x}{9 + x^2}.$$ Use calculus to find the coordinates of the turning points of \(C\). [6]
  2. Given that $$y = (1 + e^{2x})^{\frac{3}{2}},$$ find the value of \(\frac{dy}{dx}\) at \(x = \frac{1}{2} \ln 3\). [5]