? asked in 科學數學 · 1 decade ago

微積分題目 關於反函數

(A) If f is a one-one, twice differentiable function with inverse function g, shoe that

g''(x)= - {f''(g(x))/[f'(g(x))]^3}

(B) Deduce that if f is increasing and concave upward, then its inverse function is concave downward

抱歉那個第一小題我實在不知道怎麼打才會比較能夠瞭解

感謝啦各位大大!

1 Answer

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  • 1 decade ago
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    1.f(g(x))=x…………合成函數為恆等函數

    f’(g(x)) g’(x)=1…..左右微分

    f’ ’(g(x)) g’(x) g’(x)+ f’(g(x)) g’’(x)=0….再微分

    整理: g’’(x)=-[ f’ ’(g(x)) g’(x)^2]/ f’(g(x))

    2.由上

    f is increasing ……. f’(g(x))>0

    concave upward…. f’ ’(g(x))>0

    得g’’(x)<0, is concave downward

    2010-01-03 12:06:46 補充:

    1.f(g(x))=x…………合成函數為恆等函數

    f’(g(x)) g’(x)=1......g’(x)=1/f’(g(x)) ..............(!)

    f’ ’(g(x)) g’(x) g’(x)+ f’(g(x)) g’’(x)=0….再微分

    整理: g’’(x)=-[ f’ ’(g(x)) g’(x)^2]/ f’(g(x))

    =-[ f’ ’(g(x)) ]/ [f’(g(x))]^3.....................由(!)

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