SL 5.2 Connect derivative functions to increasing and decreasing behaviour | International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Calculus

IB Mathematics: Analysis and Approaches SL — study topic SL 5.2

A function is increasing on an interval when its outputs rise as the input moves from left to right; it is decreasing when its outputs fall. The derivative gives the slope of a function at each input, so its sign links directly to this behaviour. This lesson uses derivatives to identify intervals of increase and decrease, with attention to the function’s domain. You should be comfortable substituting values into expressions, solving simple equations, and reading intervals on a number line. Conclusions apply only on parts of the domain where the function is defined.

What you will learn

Explain how the sign of a derivative describes whether a function is increasing or decreasing.
Use a derivative to find intervals of increase and decrease on a stated domain.
Interpret derivative signs using algebraic, graphical, numerical, and contextual representations.
Use graphing technology to check a sign analysis without replacing the mathematical reasoning.

1. Prior knowledge: slope and derivative

For a straight line, the gradient describes how the output changes as the input increases. A positive gradient means the line rises from left to right; a negative gradient means it falls. A curve can have a different slope at different points, so its slope is described by a derivative function.

The derivative of a function

f

is written

f'(x)

. Its value at a particular input gives the slope of the graph of

f

at that input. For example, if

f'(2)=5

, the graph has positive slope at

x=2

. This tells you about one point; to describe behaviour over an interval, examine the derivative throughout that interval.

An interval such as

(1,4)

contains all values strictly between

1

and

4

. The domain is the set of inputs for which a function is defined. When you state intervals of increase or decrease, keep them inside the stated domain.

f'(x)>0\Rightarrow f is increasing

If $f'(x)>0$ throughout an interval, $f$ is increasing there.
If $f'(x)<0$ throughout an interval, $f$ is decreasing there.
A point where the derivative is zero is worth checking, but does not by itself show that the function changes direction.

2. Use the derivative sign to find intervals

Begin by stating the domain and finding

f'(x)

. Identify inputs in the domain where

f'(x)=0

or where the derivative is undefined. These values can divide the domain into intervals. On each interval, determine whether the derivative is positive or negative, using a test value or the factors of the derivative.

For example, if

f'(x)=(x-1)(x+3)

, the derivative is zero at

x=1

and

x=-3

. These values split the real number line into three intervals. Testing one value in each interval determines the sign there: positive signs mean increase, and negative signs mean decrease.

Check the sign on either side of a zero. A derivative may be positive on both sides of a point where it equals zero, so the function need not change from increasing to decreasing. A change from positive to negative means the function changes from increasing to decreasing; a change from negative to positive means it changes from decreasing to increasing.

The graph of

f

makes increasing behaviour visible as an upward trend from left to right and decreasing behaviour as a downward trend. The graph of f' provides a sign map: above the horizontal axis, its values are positive; below the axis, they are negative. These signs describe the slopes of

f

.

f'(x)<0\Rightarrow f is decreasing

Find zeros and any undefined points of f', while respecting the domain of $f$ .
Determine the sign of f' between consecutive dividing values.
Do not infer a change in behaviour from $f'(x)=0$ alone.

3. Numerical, graphical, and contextual interpretations

A table of derivative values can help check a sign analysis. Positive sampled values suggest increasing behaviour near those inputs, but a few samples do not prove that the derivative stays positive throughout an interval. Finding zeros and checking signs analytically gives the interval boundaries and the reasoning.

Graphing technology is useful for verification. After finding the derivative and its important values analytically, graph the original function or its derivative. Check whether the graph’s shape agrees with your interval conclusions. A table of values or a closer viewing window can help investigate a suspected zero. Calculator displays are approximate, however, and a viewing window may hide features; use exact boundaries when algebra provides them.

In a context, the input might be time and the function might represent distance or temperature. A positive derivative means that the measured quantity is increasing with respect to the input; a negative derivative means it is decreasing. Units help explain the rate. If distance is measured in kilometres and time in hours, the derivative has units of kilometres per hour. Conclusions must remain within the stated time interval and the model’s domain.

\operatorname{units}(f')=\frac{\operatorname{units}(f)}{\operatorname{units}(x)}

A graph or numerical table supports the analysis but does not replace a justified sign check.
Interpret the derivative using the variables, domain, and units in the question.
The units of a derivative are the units of the output divided by the units of the input.

4. A reliable solution method

State the domain first. Different domains can lead to different interval answers for the same formula. Differentiate, factor the derivative where possible, and identify its zeros and any points where it is undefined.

Make a sign chart or choose a test value from each resulting interval. Use the sign of the derivative to state where the function increases and decreases. Write intervals clearly and exclude values outside the domain. If graphing technology is available, compare your conclusions with the graph of the function or derivative.

A complete explanation normally shows the derivative, the values that divide the domain, the sign on each interval, and the resulting behaviour. A graphing-calculator observation such as “the curve goes up” can be a useful check, but it does not show how the intervals were established.

\operatorname{sign}(f'(x))\longrightarrowbehaviour of f(x)

Use the sequence: domain, derivative, dividing values, signs, behaviour.
Use exact boundaries when available and keep all intervals within the domain.
Separate analytical reasoning from a calculator check.

Derivative sign and function behaviour

Derivative sign on an interval	Slope of the function	Behaviour of the function
$f'(x)>0$	Positive	Increasing
$f'(x)<0$	Negative	Decreasing
$f'(x)=0$ at a point	Slope is zero at that point	Check signs on either side

Worked example

A quadratic that changes direction

For

f(x)=x^2-4x+1

on the real numbers, find the intervals where

f

is increasing and decreasing.

Differentiate
Differentiate each term. The resulting function gives the slope of the curve at each input.
$f'(x)=2x-4$
Find a dividing value
Set the derivative equal to zero. This identifies where the slope is zero and gives a value to check when dividing the domain.
$2x-4=0\Rightarrow x=2$
Check the signs
Choose one input on each side of $2$ . The derivative is negative at $x=0$ and positive at $x=3$ , so the function decreases before $2$ and increases after it.
$f'(0)=-4<0,\qquad f'(3)=2>0$
State the intervals
The domain is all real numbers. The negative derivative on the interval to the left of $2$ means $f$ is decreasing there; the positive derivative to the right means $f$ is increasing there.
$(-\infty,2),\qquad(2,\infty)$

Answer: The function is decreasing on

(-\infty,2)

and increasing on

(2,\infty)

.

Check: The graph is an upward-opening parabola with its lowest point at

x=2

. This agrees with the derivative changing from negative to positive.

Worked example

Factoring reveals three intervals

For

g(x)=x^3-3x

on the real numbers, determine where

g

increases and decreases.

Differentiate and factor
The derivative is quadratic. Factoring it makes its zeros easier to identify and helps with the sign check.
$g'(x)=3x^2-3=3(x-1)(x+1)$
Find dividing values
The derivative is zero at $x=-1$ and $x=1$ . These values split the real line into three intervals.
$g'(x)=0\Rightarrow x=-1,\ 1$
Determine the signs
Test $x=-2$ , $x=0$ , and $x=2$ , one from each interval. The signs are positive, negative, and positive, respectively.
$g'(-2)>0,\qquad g'(0)<0,\qquad g'(2)>0$
Report behaviour
The derivative is positive to the left of $-1$ , negative between $-1$ and $1$ , and positive to the right of $1$ . Therefore, $g$ increases on the first and third intervals and decreases on the middle interval.
$(-\infty,-1),\qquad(-1,1),\qquad(1,\infty)$

Answer: The function is increasing on

(-\infty,-1)

and

(1,\infty)

, and decreasing on

(-1,1)

.

Check: A graph of

g

should rise, then fall, then rise. The derivative sign analysis explains these changes.

Worked example

A zero derivative without a direction change

For

h(x)=x^3

on the real numbers, determine where

h

increases and decreases.

Differentiate
Find the derivative function to describe the slope at each input.
$h'(x)=3x^2$
Locate the zero
The derivative is zero at $x=0$ . Check its sign on both sides rather than assuming the function changes direction there.
$3x^2=0\Rightarrow x=0$
Check both sides
For every nonzero real input, its square is positive. Thus the derivative is positive on both intervals, even though it equals zero at the origin. h'(x)>0 for x<0 and x>0
Conclude
The function is increasing on both sides of zero. There is no interval on which it decreases.
$(-\infty,0),\qquad(0,\infty)$

Answer: The function is increasing on

(-\infty,0)

and

(0,\infty)

, and it has no interval of decrease.

Check: The graph rises from left to right through the origin. The zero derivative at the origin is consistent with a horizontal slope at that point, not a change to decreasing behaviour.

Common mistakes and how to avoid them

Assuming that

f'(x)=0

automatically means the function changes direction.

Correction: Check the derivative sign on both sides. The signs may remain the same, as they do for

h(x)=x^3

.

Using the sign of

f(x)

to decide whether the function increases or decreases.

Correction: The sign of

f'(x)

determines increasing or decreasing behaviour. The sign of

f(x)

only tells whether the function’s output is positive or negative.

Reporting behaviour outside the stated domain.

Correction: Restrict every interval to values where the original function is defined and the derivative sign analysis applies.

Using a calculator graph as the entire justification.

Correction: Find derivative zeros and check derivative signs analytically when possible. Use the graph as verification.

Lesson summary

The derivative $f'(x)$ gives the slope of $f$ at each input where the derivative exists.
Where $f'(x)>0$ , the function is increasing; where $f'(x)<0$ , it is decreasing.
Zeros or undefined points of the derivative can divide the domain into intervals to check.
A derivative equal to zero at one point does not by itself show a change in behaviour.
Graphs and numerical checks support sign analysis; the domain and context determine how to report the result.

Check your understanding

Question 1

Suppose

p'(x)=5-2x

for all real

x

. On which interval is

p

increasing?

$(-\infty,\frac{5}{2})$
$(\frac{5}{2},\infty)$
$(-\infty,5)$
All real numbers

Show answer and explanation

(-\infty,\frac{5}{2})

Solving

5-2x=0

gives

x=\frac{5}{2}

. The derivative is positive for

x<\frac{5}{2}

, so

p

is increasing on that interval.

Question 2

A function has

q'(x)<0

for every

x

in

(2,7)

. What can you conclude?

$q$ is decreasing on $(2,7)$ .
$q$ is increasing on $(2,7)$ .
$q(x)$ is negative for every $x$ in $(2,7)$ .
$q'(x)=0$ throughout $(2,7)$ .

Show answer and explanation

q

is decreasing on

(2,7)

.

A negative derivative means the slope is negative, so the function decreases on the stated interval. It does not determine whether the function’s values are positive or negative.

Question 3

If

r'(x)=x^2

on the real numbers, which statement is best supported?

$r$ increases on each side of zero; the zero derivative at zero does not show a decrease.
$r$ decreases on both sides of zero because $r'(0)=0$ .
$r$ increases for $x<0$ and decreases for $x>0$ .
$r$ must be negative for every real input.

Show answer and explanation

r

increases on each side of zero; the zero derivative at zero does not show a decrease.

Since

x^2>0

for every nonzero

x

, the derivative is positive on both sides of zero. Its value at the single point zero is not evidence of a decreasing interval.

Key terms

Derivative function: The function $f'(x)$ whose value gives the slope of $f$ at each input where the derivative exists.
Increasing: A function is increasing on an interval when its values rise as the input moves from left to right.
Decreasing: A function is decreasing on an interval when its values fall as the input moves from left to right.
Sign analysis: Checking where an expression is positive, negative, or zero, often by testing intervals separated by its zeros.

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SL 5.2 · Connect derivative functions to increasing and decreasing behaviour

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

IB Mathematics: Analysis and Approaches SL — study topic SL 5.2

What you will learn

1. Prior knowledge: slope and derivative

2. Use the derivative sign to find intervals

3. Numerical, graphical, and contextual interpretations

4. A reliable solution method

Derivative sign and function behaviour

A quadratic that changes direction

Factoring reveals three intervals

A zero derivative without a direction change

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through IB AA SL

About this lesson and its review