Absolute Minimum

College/University

Definition

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

Worked examples

\(f(x) = x^2\) has absolute minimum \(0\) at \(x = 0\)

The parabola's lowest point is at the vertex; that y-value is the absolute minimum.

\(g(x) = \sin(x)\) has absolute minimum \(-1\) at \(x = \frac{3\pi}{2}, \frac{7\pi}{2}, \ldots\)

The sine wave reaches its lowest value at infinitely many points, but the minimum value is still \(-1\).

Common mistakes

Calling the x-coordinate the absolute minimum → The absolute minimum is the y-value (output), not the x-value where it occurs The minimum is the smallest function value, not the location.
Confusing absolute minimum with local minimum → Absolute minimum is the smallest value on the entire curve; local is smallest nearby A function can have many local minima but only one absolute minimum value.
Assuming every function has an absolute minimum → \(f(x) = x\) has no absolute minimum; it decreases without bound Only functions that are bounded below on their domain can have an absolute minimum.

Where you'll use it next

You'll find absolute minima in optimization problems across calculus, use the concept when analyzing graphs in precalculus, and apply it in economics, physics, and engineering to minimize cost, energy, or error.

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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