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Math Help - Proofs

  1. #1
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    Proofs

    a) Show that, if f is convex, then -f is concave (and vice versa)

    b) Show that the following function, having domain {0,1,...,n-1}, is convex:
    f(j) = 1 / (n-j)

    I know what convex/concave looks like, but not sure what it means in a formula. Any help would be great thanks!
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  2. #2
    ynj
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    Quote Originally Posted by jaclyn91 View Post
    a) Show that, if f is convex, then -f is concave (and vice versa)

    b) Show that the following function, having domain {0,1,...,n-1}, is convex:
    f(j) = 1 / (n-j)

    I know what convex/concave looks like, but not sure what it means in a formula. Any help would be great thanks!
    convex means \forall x,y,\lambda\in [0,1],f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)
    or \forall x<c<y,\frac{f(x)-f(c)}{x-c}\leq\frac{f(x)-f(y)}{x-y}\leq\frac{f(y)-f(c)}{y-c}
    or f'(x)exists and it is non-decreasing
    or f''(x)\geq 0
    for concave, you just replace all > with <!
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