How do you find the coordinates of stationary points?

Differentiate, set \frac{dy}{dx}=0, solve for x, then substitute into y.

Second derivative test for a minimum?

\frac{d^2y}{dx^2}0 at the stationary point.

Second derivative test for a maximum?

\frac{d^2y}{dx^2}<0 at the stationary point.

What if \frac{d^2y}{dx^2}=0 at a stationary point?

The test is inconclusive; check the sign of the gradient on either side.

Gradient pattern for a local maximum?

\frac{dy}{dx} goes positive, zero, negative.

Gradient pattern for a local minimum?

\frac{dy}{dx} goes negative, zero, positive.

Condition for f to be increasing?

f'(x)0 throughout the interval.

Condition for f to be decreasing?

f'(x)<0 throughout the interval.

What is a turning point?

A stationary point where the gradient changes sign: a local maximum or minimum.

Stationary points and increasing and decreasing functionsEdexcel International A Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

What is a stationary point?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

What is a stationary point?: A point where $\frac{dy}{dx}=0$ (horizontal tangent).
How do you find the coordinates of stationary points?: Differentiate, set $\frac{dy}{dx}=0$ , solve for $x$ , then substitute into $y$ .
Second derivative test for a minimum?: $\frac{d^2y}{dx^2}>0$ at the stationary point.
Second derivative test for a maximum?: $\frac{d^2y}{dx^2}<0$ at the stationary point.
What if $\frac{d^2y}{dx^2}=0$ at a stationary point?: The test is inconclusive; check the sign of the gradient on either side.
Gradient pattern for a local maximum?: $\frac{dy}{dx}$ goes positive, zero, negative.
Gradient pattern for a local minimum?: $\frac{dy}{dx}$ goes negative, zero, positive.
Condition for $\mathrm{f}$ to be increasing?: $\mathrm{f}'(x)>0$ throughout the interval.
Condition for $\mathrm{f}$ to be decreasing?: $\mathrm{f}'(x)<0$ throughout the interval.
What is a turning point?: A stationary point where the gradient changes sign: a local maximum or minimum.
Where is $y=x^3-12x$ decreasing?: $3x^2-12<0$ , so $-2<x<2$ .
A cubic has local maximum $12$ and local minimum $-15$ . For which $k$ does cubic $=k$ have three distinct roots?: $-15<k<12$ .

Exam questions on Stationary points and increasing and decreasing functions

The curve $C$ has equation $y=x^3-6x^2+9x+2$ .
Find the $y$ -coordinates of the two stationary points of $C$ .2 marks
The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=2x^3-3x^2-12x+5$ .
Hence find the range of values of $k$ for which the equation $\mathrm{f}(x)=k$ has three distinct real roots.2 marks
The curve $C$ has equation $y=x+\frac{4}{x^2}$ , for $x>0$ .
Find $\frac{dy}{dx}$ and hence find the coordinates of the stationary point of $C$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).