5 · IB Maths AA
Second derivative and points of inflexion
Graph f, f′ and f″ together and drag the point along the curve to see where f is concave up or down, where the stationary points are and where the points of inflexion lie.
f f′ f″ concave up concave down · ● stationary point · ◆ point of inflexion · ○ f″ = 0 only
Try this
- With f(x) = x³ − 3x, drag from left to right and note where the shading changes; check that the green f″ line crosses the x-axis at exactly that x.
- Choose f(x) = ¼x⁴ − x³. At x = 0 both f′ and f″ are zero; compare this with x = 2, where only f″ is zero. What kind of point is each?
- Choose f(x) = x⁴. At x = 0 the readout shows f″ = 0, yet the shading does not change. Explain why this is a minimum and not a point of inflexion.
Common error: showing f″(a) = 0 is not enough to claim a point of inflexion — you must also show that f″ changes sign either side of x = a, as x⁴ at x = 0 demonstrates.