The First Derivative and Function Behavior

Calculus 1 · Axiom Academy

LESSON The First Derivative and Function Behavior How the sign of f (x) tells you exactly where a function rises, falls, and turns around. 1. The Slope's Sign Decides the Direction Slide a point along the curve f(x) = x^3 - 3x^2 - 9x + 5 and keep a tangent line glued to it. Watch the line's tilt : it points up while the function rises, goes flat the instant it turns around, then points down as the function falls. That tilt is exactly f'(x) — so the sign of the derivative is the direction of travel. Positive slope → the function is increasing Negative slope → the function is decreasing The direction can only flip where the slope passes through zero. So the first concrete step is to differentiate and solve f'(x) = 0 . For our cubic the derivative is a parabola, and the two places it crosses the axis are the critical points — the only candidates for a peak or valley. Power rule term by term: f'(x) = 3x^2 - 6x - 9 . Pull out 3, then factor the quadratic: 3(x+1)(x-3) . A product is zero when a factor is: x+1=0 or x-3=0 . x = -1 and x = 3 . The slope is zero at exactly these two x -values. The two critical points x=-1 and x=3 split the number line into three intervals. Inside each interval the sign of f' cannot change, so we only need to test one point per interval . Plug each test value into f'(x)=3(x+1)(x-3) and record whether the result is + or - . 4. Read the Intervals & Classify the Extrema

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