From the graph of f to the graph of f′
Find where f′ is zero, read f′ from the graph of f, locate the point of inflection as an extremum of f′ and the moment of fastest growth.
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Guide
The derivative f′(x₀) is the slope of the tangent to the graph of f at x₀. So f′ is large where the graph is steep and f′ = 0 where the tangent is horizontal, at a high point or a low point.
Where the graph of f rises, f′ is positive. Where it falls, f′ is negative. At a maximum f′ changes from + to −, at a minimum from − to +.
The graph of f is steepest where its slope f′ has an extremum: at a point of inflection (Wendepunkt). If f is an amount that grows over time, the Wendestelle is the moment of the greatest growth rate.