mylm 發問時間: 科學數學 · 1 0 年前

微積分1題

Suppose f is differentiable on (a,b),continuous on[a,b],and

f(a)=f(b)=0. Prove that if f(c)>0 for some c∈(a,b),then there exist

x1,x2 ∈(a,b) such thar f'(x1)>> f'(x2).

1 個解答

評分
  • 1 0 年前
    最佳解答

    Applying the Mean Value Theorem:

    There exists x1 ∈ (a, c) such that:

    f'(x1) = [f(c) - f(a)]/(c - a)

    = f(c)/(c - a)

    > 0 since c > a

    Also, there exists x2 ∈ (c, b) such that:

    f'(x2) = [f(b) - f(c)]/(b - c)

    = -f(c)/(b - c)

    < 0 since b > c

    Hence f'(x1) > f'(x2) with x1, x2 ∈ (a, b)

    參考資料: Myself
還有問題?馬上發問,尋求解答。