CAIE A-Level Mathematics AS 1.7.4 Stationary Points Questions

A stationary point occurs where the curve’s gradient is zero. Solve f′(x)=0 for its coordinates, then use f′′ or nearby gradient signs to determine its nature.

Exam points

Find stationary points by solving f′(x)=0, including equations with unknown constants, and give the coordinates when required.
Classify stationary points as maxima or minima using the second derivative or sign changes in the first derivative; establish when no other stationary points exist and evaluate the relevant extremum when asked.

[Maximum number: 5]

The equation of a curve is $y=2 x^{2}-\frac{1}{2 x}+3$ .

(a)

Find the coordinates of the stationary point.

[ 3 ]

Differentiate to obtain $4 x+\frac{1}{2} x^{-2}$
OE
Condone ' +c '.

Equate first derivative to zero and solve $4 x+\frac{K}{x^{2}}=0$ as far as $x^{3}=k, K$ and k nonzero
Equate first derivative to zero and solve $4 x+\frac{K}{x^{2}}=0$ as far as $x^{3}=k, K$ and k non-
zero
Not given if ' +c ' used.

$x=-\frac{1}{2}$ and $y=\frac{9}{2}$
OE
B1 SC if no visible solution of the cubic.

(b)

Determine the nature of the stationary point.

[ 2 ]

Differentiate their first derivative, substitute their x value. Substitution may be
implied by a correct inequality or correct value,
Must differentiate one term correctly.

Expect $4-x^{-3}=12$ at $x=\frac{-1}{2}$

Alternative: substitute values of x into $\frac{\mathrm{d} y}{\mathrm{~d} x}$ .

One value $x<-\frac{1}{2}$ and one value $-\frac{1}{2}<x<0$ .

conclude minimum
Following correct work only

All question bank results loaded

CAIE A-Level Mathematics AS 1.7.4 Stationary Points Questions question 1

Question (a)

Question (b)