Higher Mathematics

Higher Maths Maximum and minimum on closed intervals

Practise maximum and minimum on closed intervals for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Compare stationary values with endpoint values to find greatest and least values on a closed interval.

Topic practice Formula support Mixed revision

Before you start

Find stationary points.
Evaluate a function accurately.
Understand closed interval notation.

Higher Mathematics lesson

Explanation

The greatest and least values on a closed interval may occur at stationary points or at endpoints.

List every candidate x-value, evaluate the original function at each, then compare the outputs.

Method and rules

Candidates are endpoints and interior points where f'(x) = 0
Evaluate f, not f', at every candidate.
State the greatest or least function value and where it occurs.

Worked examples

Worked example 1

Check endpoints and stationary points

Find the greatest and least values of f(x) = x² − 4x + 1 on 0 ≤ x ≤ 5.

Solve f'(x) = 2x − 4 = 0, giving x = 2.
Evaluate f(0), f(2) and f(5)
Compare 1, −3 and 6.

Answer: Least value −3 at x = 2; greatest value 6 at x = 5

Worked example 2

Method check

Which x-values must be tested on a closed interval?

Extrema can occur at the boundary.

So: Both endpoints and every stationary point inside the interval.

Watch out

Checking stationary points but not endpoints.
Comparing derivative values instead of function values.
Including stationary points outside the interval.

Exam reminder

For maximum and minimum on closed intervals, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.

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

Stationary points and optimisation Increasing, decreasing and derivative graphs Definite integrals and area All Higher Maths topics Higher Maths worksheets