Stationary points and turning points

Maximum and minimum turning points and stationary points of inflection, local maxima and minima, classifying stationary points with a table of slopes, and finding unknown coefficients from a stationary point.

Three kinds of stationary point

Kind	Slope just left	Slope just right
Maximum turning point	$+$	$-$
Minimum turning point	$-$	$+$
Stationary point of inflection	same sign on both sides

A maximum turning point is a local maximum: the curve is higher there than at nearby points, though it may be higher somewhere else.

Classifying with a table of slopes

Example 1●●

Find and classify the stationary points of $y = x^3 - 3x^2 - 9x + 5$.

$y' = 3(x - 3)(x + 1)$, which is zero at $x = -1$ and $x = 3$. Then $y(-1) = 10$ and $y(3) = -22$.

$x$	$-2$	$-1$	$0$	$3$	$4$
$y'$	$15$	$0$	$-9$	$0$	$15$
shape	/	—	\	—	/

Test x = −2, 0 and 4.

Maximum turning point $(-1, 10)$ and minimum turning point $(3, -22)$.

Example 2: a stationary inflection●●●

Find and classify the stationary points of $y = x^4 - 4x^3$.

$y' = 4x^3 - 12x^2 = 4x^2(x - 3)$, which is zero at $x = 0$ and $x = 3$.

Test values: $y'(-1) = -16$, $\;y'(1) = -8$, $\;y'(4) = 64$.

$(0, 0)$ is a stationary point of inflection, and $(3, -27)$ is a minimum turning point.

The slope is negative on both sides of x = 0.

y = x⁴ − 4x³

The curve flattens at the origin but keeps going down: that is a stationary point of inflection. It turns at (3, −27).

Try this

Why can't a double zero of $y'$ such as $x^2$ give a turning point?

Finding unknown coefficients

Example 3●●●

The curve $y = x^3 + ax^2 + b$ has a stationary point at $(2, -3)$. Find $a$ and $b$.

$y' = 3x^2 + 2ax$, and $y'(2) = 0$ gives $12 + 4a = 0$, so $a = -3$.

The point is on the curve: $8 - 12 + b = -3$, so $b = 1$.

Flashcards

Maximum turning point

Slope $+$, then $0$, then $-$.

Stationary point of inflection

Slope $0$ with the same sign on both sides.

Table of slopes

Test the sign of $f'$ either side of each stationary point.

Practice

Find the $y$-coordinate of the maximum turning point of $y = x^3 - 3x^2 - 9x + 5$.
Find the $y$-coordinate of its minimum turning point.
For $y = x^4 - 4x^3$, the stationary point at the origin is:
$y = x^3 + ax^2 + b$ has a stationary point at $(2, -3)$. Find $a$.
For the same curve, find $b$.
A table of slopes shows $+$, $0$, $-$. The point is:
Find the $y$-coordinate of the maximum turning point of $y = 2x^3 - 3x^2 - 12x$.
The curve $y = x^3$ at the origin has:
A local maximum is: