Finding stationary points after integration

A question is this type if and only if it requires finding f(x) by integration using substitution, then finding and classifying stationary points or determining where f is increasing/decreasing.

3 questions · Moderate -0.1

Sort by: Default | Easiest first | Hardest first
CAIE P1 2019 November Q9
10 marks Moderate -0.3
9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).
  1. Find the equation of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. Find the coordinates of the stationary point on the curve and, showing all necessary working, determine the nature of this stationary point.
CAIE P1 2019 November Q8
8 marks Moderate -0.3
8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).
  1. Find the set of values of \(x\) for which f is decreasing.
  2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). Find \(\mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2011 June Q6
8 marks Standard +0.3
  1. Using the substitution \(u = x^2\), or otherwise, find the numerical value of $$\int_0^{\sqrt{\ln 4}} xe^{-\frac{1}{2}x^2} \, dx.$$ [4]
  2. Determine the exact coordinates of the stationary points of the curve \(y = xe^{-\frac{1}{2}x^2}\). [4]