Justify maximum/minimum value

Prove that a stationary point is a maximum or minimum using second derivative or contextual reasoning.

2 questions

SPS SPS SM Pure 2021 June Q15
15. A curve has equation \(y = \mathrm { g } ( x )\). Given that
  • \(\mathrm { g } ( x )\) is a cubic expression in which the coefficient of \(x ^ { 3 }\) is equal to the coefficient of \(x\)
  • the curve with equation \(y = \mathrm { g } ( x )\) passes through the origin
  • the curve with equation \(y = \mathrm { g } ( x )\) has a stationary point at \(( 2,9 )\)
    1. find \(g ( x )\),
    2. prove that the stationary point at \(( 2,9 )\) is a maximum.
      [0pt] [BLANK PAGE]
      [0pt] [BLANK PAGE]
AQA AS Paper 1 2018 June Q10
2 marks
10 A curve has equation \(y = 2 x ^ { 2 } - 8 x \sqrt { x } + 8 x + 1\) for \(x \geq 0\) 10
  1. Prove that the curve has a maximum point at ( 1,3 )
    Fully justify your answer.
    10
  2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]