Higher Mathematics

Higher Maths Increasing, decreasing and derivative graphs

Practise increasing, decreasing and derivative graphs for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Use the sign and graph of a derivative to describe where a function rises, falls or is stationary.

Topic practiceFormula supportMixed revision

Before you start

  • Solve factored inequalities.
  • Read whether a graph is above or below an axis.
  • Find stationary x-values.
Higher Mathematics lesson

Explanation

A function is strictly increasing where f'(x) > 0 and strictly decreasing where f'(x) < 0.

On a graph of f', crossings of the x-axis locate stationary points of f; the sign change identifies their nature.

Visual support

Derivative sign and increasing or decreasing behaviourThe derivative crosses the x-axis at negative 2 and 2. It is positive to the left of negative 2 and to the right of 2, so the original function is increasing there. It is negative between the roots, so the original function is decreasing there.x = -2x = 2xf′(x)f′ > 0increasingf′ < 0decreasingf′ > 0increasing+ to −: local maximum of f · − to +: local minimum of f

f'(x) = (x + 2)(x − 2): positive outside the roots and negative between them

Method and rules

  • f'(x) > 0: f is strictly increasing
  • f'(x) < 0: f is strictly decreasing
  • A change + to − gives a local maximum; − to + gives a local minimum.

Worked examples

Worked example 1

Use the sign of a derivative

Given f'(x) = 3(x − 1)(x + 2), state where f is increasing

  1. The derivative is zero at x = −2 and x = 1.
  2. Test the sign on the three intervals.
  3. The product is positive outside the roots.

Answer: f is strictly increasing for x < −2 and x > 1

Worked example 2

Method check

If f'(x) changes from negative to positive, what happens to f?

  1. The function changes from decreasing to increasing.

So: f has a local minimum.

Watch out

  • Solving f(x) > 0 instead of f'(x) > 0
  • Including stationary endpoints in a strictly increasing interval.
  • Reading the graph of f' as though it were f.

Exam reminder

For increasing decreasing and derivative graphs, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.

Related Higher Maths topics

Continue with the methods that connect most closely to this topic.

Stationary points and optimisationSketching polynomialsMaximum and minimum on closed intervalsAll Higher Maths topicsHigher Maths worksheets