asked in 科學數學 · 1 decade ago


Let i be an open interval of real numbers and suppose that the function

f:I→R is continuous. Let c be a real number. Fix a number x0 in the interval I and define the auxiliary function H:R→R by

H(x)=cx-∫ 上限X 下限X0 f(s)ds for x in I

For a point x in I. show that f(x)=c if H`(x)=0.

conclude that c is the image of f:I→R provided that the function

H:I→R has a local extreme point.


1 Answer

  • 1 decade ago
    Favorite Answer

    H(x)=cx-∫ (x0->x) f(s)ds

    H'(x)=c-f(x) ,so H'(x)=0=>f(x) = c

    Since if H:I→R has a local extreme point, then the critical point M should satisfied H'(M) = 0 => f(M) = c and thus c is the image of f:I→R

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