First Derivative Test

Business Calculus · Axiom Academy

Classifying critical points using sign changes The sign of the derivative tells us whether a function is increasing or decreasing: At a local maximum, the function must go UP then DOWN. At a local minimum, it must go DOWN then UP. We can detect this by checking if the derivative changes sign! Suppose c is a critical point of f (where f'(c) = 0 or f'(c) is undefined): f' changes from positive to negative at c f' changes from negative to positive at c Step 2: Find critical points (set f'(x) = 0) x = -1: Local maximum (f changes from + to −) f(-1) = -1 - 3 + 9 + 5 = 10 x = 3: Local minimum (f changes from − to +) f(3) = 27 - 27 - 27 + 5 = -22 Step 3: Apply the First Derivative Test Since R'(x) changes from positive to negative at x = 120, this is a maximum. 1. Find f'(x) 2. Find critical points (where f'(x) = 0 or undefined) 3. Create a sign chart with critical points 4. Test a value in each interval 5. Classify based on sign changes The First Derivative Test captures the fundamental geometry: a maximum is where a function stops increasing and starts decreasing, and a minimum is where it stops decreasing and starts increasing. The derivative's sign tells us exactly that!

This is the written version of the interactive lesson above. See the full Business Calculus course.