Maxima & Minima

Maths: Edexcel IGCSE Higher, topic 3.5.2

Turning Points of Graphs

The turning point of a graph is any coordinate on a graph where the gradient = 0. Another way to say this is that the derivative of the graph = 0 or that the gradient of the graph = 0.

Turning points

Once you have found the derivative of a graph, f'( $x$ ), you can find the $x$ coordinate of a turning point by making f'( $x$ ) = 0.
When f'( $x$ ) = 0, the rate of change of the function is equal to zero.
It is called a turning point because the graph often changes direction at f'( $x$ ) = 0.

Example

If f( $x$ ) = $x$ ² − 2 $x$
f'( $x$ ) = 2 $x$ − 2
The turning point is when f'( $x$ ) = 0
So, 0 = 2 $x$ − 2
Solve for $x$ to find $x$ = 1

Example cont.

Find the $y$ -coordinate by substituting in the value of $x$ = 1 into f( $x$ ).
f(1) = 1² − (2 × 1)
f(1) = −1
So the coordinate of the turning point of f( $x$ ) = $x$ ² − 2 $x$ is (1, −1).

Maxima & Minima

You can categorise turning points as a maximum or a minimum. Maxima and minima are the plurals of maximum and minimum.

Maxima & minima

A turning point is a local maximum if it is a high point when the graph is turning.
A turning point is a local minimum if it is a low point when the graph is turning.

The second derivative

You can find out if a turning point is a local maximum or local minimum using the second derivative.
The second derivative is found by differentiating the derivative.
We can write the second derivative of f( $x$ ) is f''( $x$ )
We can write the second derivative of $y$ as $\large\frac{d^2y}{dx^2}$ or $y$ ''.

Local maximum or local minimum

Once you have the $x$ value of the turning point, you can find out if it's a local maximum or a local minimum by substituting the $x$ value into the second derivative.
If the second derivative is less than 0, then it is a local maximum.
If the second derivative is greater than 0, then it is a local minimum.

Example

f( $x$ ) = $x$ ³ + 6 $x$ ² + 12 has a turning point at (0, 12). Is this a maximum or a minimum?
f'( $x$ ) = 3 $x$ ² + 12 $x$
And f''( $x$ ) = 6 $x$ + 12
Then substitute in the $x$ coordinate of the turning point into f''( $x$ ).
f''(0) = (6 × 0) + 12 = 12
f''( $x$ ) is positive at $x$ = 0 so the turning point at (0, 12) is a local minimum.

Test yourself on Maxima & Minima

How do you find the second derivative of a function?
What will the value of the second derivative be at a turning point if the turning point is a local maximum?
What will the value of the second derivative be at a turning point if the turning point is a local minimum?

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