Absolute Maximum

College/University

Definition

The largest value of a function over its entire curve. There is only one absolute maximum, but the maximum can happen in more than one place over the curve. The absolute maximum is also sometimes known as the global maximum.

Worked examples

\(f(x) = -x^2 + 4\) has absolute maximum \(4\) at \(x = 0\)

The parabola opens down, so its highest point is the vertex at the top.

\(g(x) = \sin(x)\) has absolute maximum \(1\) at \(x = \frac{\pi}{2}, \frac{5\pi}{2}, \ldots\)

The sine wave repeats, so the absolute maximum occurs at infinitely many points.

Common mistakes

Every local maximum is an absolute maximum → Only the highest local maximum is the absolute maximum A function can have many peaks, but only the tallest one is the absolute maximum.
\(f(x) = x^2\) has an absolute maximum → \(f(x) = x^2\) has no absolute maximum (grows without bound) Parabolas opening upward have absolute minimums, not maximums.
If the absolute maximum is \(5\), there is only one \(x\) where \(f(x) = 5\) → The absolute maximum value can occur at multiple \(x\)-values The maximum value is unique, but it may happen at more than one point on the curve.

Where you'll use it next

Absolute maximum is central to optimization problems in calculus, where you'll find extreme values on closed intervals, apply the Extreme Value Theorem, and solve real-world max/min applications in economics, engineering, and physics.

Found in 1 StudyPug lesson

Critical Numbers: The Key to Maximum and Minimum Values

Calculus 1

Unlock the power of critical numbers in calculus. Learn to identify key points, analyze function behavior, and solve real-world optimization problems with confidence and precision.

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