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Stationary points: f(x), f′(x), f″(x) and increasing/decreasing intervals

Compare y = f(x) directly above its derivative y = f′(x) to see how roots of f′(x) = 0 give stationary points, how the sign of f″(x) classifies them, and how to sketch y = f(x) from y = f′(x).

Function y = f(x) (green shaded = increasing f′(x) > 0, amber = decreasing f′(x) < 0)

Derivative graph y = f′(x) (roots where f′(x) = 0 align with stationary points above)

Adjust curve parameters

Stationary point x₁ (first root of f′(x) = 0)x₁ = −1.0

Stationary point x₂ (second root of f′(x) = 0)x₂ = 2.0

Leading scale factor k (flips orientation when negative)k = 0.5

Vertical shift constant +CC = 1.0

Stationary points: f(x), f′(x), f″(x) and increasing/decreasing intervals

Function y = f(x) (green shaded = increasing f′(x) > 0, amber = decreasing f′(x) < 0)

Derivative graph y = f′(x) (roots where f′(x) = 0 align with stationary points above)

Adjust curve parameters

Stationary point classification using f′(x) = 0 and f″(x)

Algebraic derivatives & interval sign analysis