Calculus Applications of differentiation

Increasing and decreasing functions

Q: How do you know if a function is increasing or decreasing?

Check the sign of f prime of x: where the derivative is positive the function is increasing, where it is negative it is decreasing.

Q: What is a critical point?

A value of x where the derivative is 0 or undefined. Critical points are the only candidates for local maxima and minima.

Q: How do you make a sign chart for the derivative?

Mark the critical points on a number line, then test one x in each interval to record whether the derivative is positive or negative there.

Q: Does f'=0 always mean a maximum or minimum?

No. The derivative equals 0 gives a critical point, but it is a turning point only if the derivative actually changes sign there.

20 practice questions 0 video lessons Theory + worked examples

Increasing and Decreasing Functions

California Calculus • Standard 9.0 • Applications of Differentiation

Increasing and Decreasing Functions is a topic in Applications of Differentiation in the California Calculus Standards. It is aligned to Standard 9.0, which requires students to use differentiation to sketch graphs by hand, here using the sign of the first derivative to find where a function increases or decreases.

A function increases where its derivative is positive (\(f'(x)>0\)) and decreases where it is negative (\(f'(x)<0\)). The critical points where \(f'=0\) separate these intervals.

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Theory

The sign of the first derivative tells you where a function rises or falls: \(f'>0\) means increasing and \(f'<0\) means decreasing. The critical points where \(f'=0\) separate these intervals.

The first derivative measures the slope, so its sign tells you the direction of the graph:

\(f'(x)>0\) on an interval \(\Rightarrow\) \(f\) is increasing there;
\(f'(x)<0\) \(\Rightarrow\) \(f\) is decreasing;
\(f'(x)=0\) marks a critical point — a possible turning point.

A sign chart of \(f'\) organizes this: mark the critical points on a number line and record the sign of \(f'\) on each interval between them.

Key idea: factor \(f'(x)\) so its sign is easy to read. Test one value in each interval, or use the factored form directly.

Rising where \(f'>0\) (teal), falling where \(f'<0\) (red).

A sign chart of \(f'\): plus for increasing, minus for decreasing.

The increasing/decreasing test:

\[f'(x)>0\ \Rightarrow\ \text{increasing},\qquad f'(x)<0\ \Rightarrow\ \text{decreasing}\]

derivative positive means increasing, negative means decreasing

Critical points solve

\[f'(x)=0\quad(\text{or } f' \text{ undefined}).\]

Turning points can only happen at critical points, but not every critical point is a turn — check the sign on both sides.

How to find increasing/decreasing intervals

Differentiate and factor \(f'(x)\).
Solve \(f'(x)=0\) for the critical points.
Test the sign of \(f'\) on each interval; \(+\) is increasing, \(-\) is decreasing.

Example 1 — Increasing or decreasing

On what intervals is \(f(x)=x^2-4x\) increasing?

Solution

Find \(f'\) and its sign.

\(f'(x)\)	\(=\)	\(2x-4\)
\(f'(x)>0\)	\(\Rightarrow\)	\(x>2\)

Increasing on \((2,\infty)\), decreasing on \((-\infty,2)\).

increasing for x greater than 2

Example 2 — A cubic

Where is \(f(x)=x^3-3x\) increasing?

Solution

Factor \(f'\) to read its sign.

\(f'(x)\)

\(=\)

\(3x^2-3=3(x-1)(x+1)\)

\(f'>0\) for \(x<-1\) and \(x>1\) (increasing); \(f'<0\) on \((-1,1)\) (decreasing).

increasing on x less than minus one and x greater than one

Example 3 — Critical points

Find the critical points of \(f(x)=x^3-6x^2+9x\).

Solution

Critical points are where \(f'(x)=0\).

\(f'(x)\)	\(=\)	\(3x^2-12x+9\)
	\(=\)	\(3(x-1)(x-3)\)
\(x\)	\(=\)	\(1,\ 3\)

critical points at x equals 1 and 3

Example 4 — Sign of \(f'\) at a point

Is \(f(x)=x^2-6x\) increasing or decreasing at \(x=1\)?

Solution

Check the sign of the derivative there.

\(f'(x)\)	\(=\)	\(2x-6\)
\(f'(1)\)	\(=\)	\(-4<0\)

Since \(f'(1)<0\), \(f\) is decreasing at \(x=1\).

decreasing at x equals 1

Common pitfalls

Use \(f'\), not \(f\). The sign of the derivative controls increase/decrease, not the sign of the function itself.

Critical points are not automatically extrema. \(f'\) must change sign there for a turning point.

Include where \(f'\) is undefined. A cusp or vertical tangent can also separate increasing and decreasing pieces.

Frequently asked questions

How do you know if a function is increasing or decreasing?

Check the sign of \(f'(x)\): where \(f'>0\) the function is increasing, where \(f'<0\) it is decreasing.

What is a critical point?

A value of \(x\) where \(f'(x)=0\) or \(f'\) is undefined. Critical points are the only candidates for local maxima and minima.

How do you make a sign chart for the derivative?

Mark the critical points on a number line, then test one \(x\) in each interval to record whether \(f'\) is \(+\) or \(-\) there.

Does f'=0 always mean a maximum or minimum?

No. \(f'=0\) gives a critical point, but it is a turning point only if \(f'\) actually changes sign there.

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Local maxima and minima

Increasing and decreasing functions

Increasing and Decreasing Functions

📖 Theory

How to find increasing/decreasing intervals

Common pitfalls

Frequently asked questions

Theory