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Applications of the First Derivative
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Increasing and decreasing intervals, stationary points (maxima and minima), and optimisation
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Stationary points & Increasing/Decreasing Regions
SL 5.2
f′(x)⎩⎪⎨⎪⎧<0⇔f decreasing=0⇔f stationary>0⇔f increasing🚫
Maxima & Minima
SL AI 5.6
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.
Optimisation
SL AI 5.7
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.
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