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Math Help - Existence of a root

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    Existence of a root

    We have two functions f,g : [a,b] \longrightarrow \mathbb{R} such that g is continuous, f+g is decreasing, f(a) < 0 and f(b) > 0. Prove that there exists a < c < b such that f(c) = 0.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by NonCommAlg View Post
    We have two functions f,g : [a,b] \longrightarrow \mathbb{R} such that g is continuous, f+g is decreasing, f(a) < 0 and f(b) > 0. Prove that there exists a < c < b such that f(c) = 0.
    Binary search will converge to a point c such that for all 0<\varepsilon<\varepsilon_0 (that is for all \varepsilon small enough):

     <br />
f(c-\varepsilon)<0 and f(c+\varepsilon)\ge 0<br />

    Thus either f(c)=0 or there is an increasing jump discontinuity at c. But an increasing jump at c is imposible given that g(x) is continuous and f(x)+g(x) and decreasing at c.

    CB
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