Stationary point via first principles

A question is this type if and only if it asks students to show that a point is stationary by finding the gradient of chord AB in terms of h and explaining why this shows the gradient is zero.

3 questions · Moderate -0.3

1.07a Derivative as gradient: of tangent to curve
Sort by: Default | Easiest first | Hardest first
AQA FP1 2010 June Q5
6 marks Moderate -0.8
5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).
  1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
  2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
    \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}
AQA FP1 2013 June Q5
8 marks Standard +0.3
5
  1. A curve has equation \(y = 2 x ^ { 2 } - 5 x\).
    The point \(P\) on the curve has coordinates \(( 1 , - 3 )\).
    The point \(Q\) on the curve has \(x\)-coordinate \(1 + h\).
    1. Show that the gradient of the line \(P Q\) is \(2 h - 1\).
    2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point \(P\) is parallel to the line \(x + y = 0\).
  2. For the improper integral \(\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x\), either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.
AQA Paper 1 2018 June Q15
6 marks Moderate -0.5
15 A curve has equation \(y = x ^ { 3 } - 48 x\) The point \(A\) on the curve has \(x\) coordinate - 4
The point \(B\) on the curve has \(x\) coordinate \(- 4 + h\) 15
  1. Show that the gradient of the line \(A B\) is \(h ^ { 2 } - 12 h\) 15
  2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-25_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-26_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-27_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-28_2498_1721_213_150}