嚴(yán)格意義上：函數(shù)的單調(diào)增減性和函數(shù)一階導(dǎo)函數(shù)的正負(fù)有關(guān),函數(shù)在某區(qū)間遞增的充要條件是該函數(shù)的導(dǎo)函數(shù)在該區(qū)間為正；反之,函數(shù)在某區(qū)間遞減的充要條件是該函數(shù)的導(dǎo)函數(shù)在該區(qū)間為負(fù).
然而函數(shù)的二階導(dǎo)函數(shù)不能判斷函數(shù)的單調(diào)性.同理把一階導(dǎo)函數(shù)當(dāng)作函數(shù),則二階導(dǎo)函數(shù)是一階導(dǎo)函數(shù)的導(dǎo)函數(shù),所以二階導(dǎo)函數(shù)能判斷一階導(dǎo)函數(shù)的單調(diào)性.然而,二階導(dǎo)函數(shù)可以判斷函數(shù)的極大值和極小值,這要在一階導(dǎo)函數(shù)取值為0的點(diǎn)處判斷,若在該點(diǎn)二階導(dǎo)函數(shù)大于零,則改點(diǎn)函數(shù)值為函數(shù)的極小值；反之,是函數(shù)的極大值.