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Maxima and minima
Critical points and the second-derivative test.
A smooth function peaks or bottoms out only where its derivative is zero. Find those critical points, then use the second derivative to tell a maximum (f″ < 0) from a minimum (f″ > 0). This is the whole of optimisation in one variable.
Beispiel: critical points of x^3 - 3x
Schritt für Schritt
- f(x) = x^{3} - 3 x
Critical points are where f′(x) = 0 or is undefined.
- f'(x) = 3 x^{2} - 3
Differentiate.
- x = -1, x = 1
Solve f′(x) = 0.
- f''(x) = 6 x
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
- f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}
- f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}
Die Antwort aufzeigen
(-1, 2)\ \text{local maximum},\; (1, -2)\ \text{local minimum}
Versuch es selbst.
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LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sums