Why set the derivative to zero?

At a maximum or minimum the tangent line is horizontal, so the instantaneous rate of change — the derivative — is zero there.

How do you know if a critical point is a max or a min?

Use the second derivative test: negative means a maximum, positive means a minimum; or check the sign of the first derivative on each side.

Grade 12 · Mathematics

Optimization

Why it matters

Max and min problems.

Advanced20m readingPrerequisite: Related rates

How do you solve optimization problems in calculus?

Short answer

Optimization finds the maximum or minimum value of a quantity. You express the quantity as a function of one variable, differentiate it, and set the derivative equal to zero to find critical points, since extrema occur where the slope is zero (or at endpoints). The second derivative or a sign test confirms whether each critical point is a maximum or a minimum.

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Frequently asked questions

Why set the derivative to zero?: At a maximum or minimum the tangent line is horizontal, so the instantaneous rate of change — the derivative — is zero there.
How do you know if a critical point is a max or a min?: Use the second derivative test: negative means a maximum, positive means a minimum; or check the sign of the first derivative on each side.

How do you solve optimization problems in calculus?

Related topics

Frequently asked questions