The sign of the derivative and the shape of the graph
Use the sign of f′(x) to find where a graph rises and falls, place the stationary points, find the point of inflection with f″(x) and work out a maximum value.
- UK: Year 12
- France: grade 11
- Germany: grade 11
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 11
- Palestine: grade 12
- Ukraine: grade 10
- Russia: grade 11
Guide
If f′(x) > 0 the graph is going up: f is an increasing function. If f′(x) < 0 the graph is going down: f is a decreasing function.
Where f′(x) = 0 the tangent is horizontal: a stationary point. If the sign of f′ changes from positive to negative it is a maximum (a peak); from negative to positive it is a minimum (a valley).
f″(x) tells you which way the graph bends. Where the sign of f″ changes there is a point of inflection. At a minimum f″(x) > 0 and at a maximum f″(x) < 0.