✏️ Разбор задачи с экзамена ШАД
Условие: Непрерывная функция f(x) такова, что f(0)=f(2). Докажите, что для какого-то x ∈ [0,2] имеет место равенство f(x) = f(x-1).
Решение: Рассмотрим функцию
φ(1) = f(1) - f(0)
φ(2) = f(2) - f(1)
В силу равенства $f(0)=f(2)$ из условия, получим φ(1)=-φ(2). Функция φ(x) непрерывна на отрезке [1, 2], тогда по теореме Больцано-Коши о промежуточном значении найдется
Ответ:
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