Izrek o povprečni vrednosti

Za vsako funkcijo, ki je zvezna na [a, b] in odvedljiva na (a, b), obstaja neka točka c na odprtem intervalu (a, b), da je sekanta, ki povezuje obe končni točki intervala [a, b], vzporedna tangenti v c.

Izrèk o povpréčni vrédnosti (tudi Lagrangeev izrèk ali izrèk o kônčnem prirástku fúnkcije) je v matematični analizi izrek, ki pravi, da v danem odseku gladke krivulje obstaja točka, v kateri je odvod (nagib) krivulje enak »povprečnemu« odvodu intervala. Izrek se uporablja pri dokazovanju izrekov, ki obravnavajo funkcije na intervalu.

Povprečna vrednost integrabilne funkcije na intervalu je število:

Izrek lahko razumemo tudi s pomočjo gibanja. Če avtomobil prepotuje v eni uri 100 km in je njegova povprečna hitrost enaka 100 km/h, potem je morala biti v nekem trenutku njegova trenutna hitrost enaka natanko 100 km/h.

Izrek je prvi razvil Joseph-Louis de Lagrange in se imenuje tudi po njem.

Formalna izjava

Naj je funkcija v zaprtem intervalu zvezna in v odprtem intervalu odvedljiva. Tedaj obstaja vsaj eno takšno število med a in b, da je:

Če pišemo drugače, in označimo s število med 0 in 1:

Posplošitev Lagrangeevega izreka je Taylorjev izrek.

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