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Applications of the First Derivative
Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation
Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation
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Stationary points are often local extrema.
If ​f′(a)=0, ​f​ is decreasing to the left of ​a​ (​f′(x)<0​), and ​f​ is increasing to the right of ​a​ (​f′(x)>0​), then ​(a,f(a))​ is a local minimum.
If ​f′(a)=0, ​f​ is increasing to the left of ​a​ (​f′(x)<0​), and ​f​ is decreasing to the right of ​a​ (​f′(x)>0​), then ​(a,f(a))​ is a local maximum.
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Optimisation problems require you to find a minimum or maximum value by producing a function ​f(x), taking its derivative, solving ​f′(x)=0, and confirming which stationary point(s) are minima or maxima.