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Math Help - Real Analysis

  1. #1
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    Real Analysis

    Let f:[0,2] -> R be continuous on [0,2] and differentiable on (0,2) with f(x)=0 and f(1)=f(2)=1

    Show that there is a c(1) in (0,1) with f (c(1)) = 1

    Show that there is a c(2) in (1,2) with f (c(2)) = 0

    Show that there is a c(3) in (0,2) with f (c(3)) = 1/3
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  2. #2
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    Quote Originally Posted by dhhnerd View Post
    Let f:[0,2] -> R be continuous on [0,2] and differentiable on (0,2) with f(x)=0 and f(1)=f(2)=1

    Show that there is a c(1) in (0,1) with f (c(1)) = 1

    Show that there is a c(2) in (1,2) with f (c(2)) = 0

    Show that there is a c(3) in (0,2) with f (c(3)) = 1/3
    These are just applications of the Mean Value Theorem, I think for #3 you want to say f(c_3) = 1/2.

    In that case just by MVT we get:
    1) f(1)-f(0)=f'(c_1)(1-0)\implies f'(c_1)=1
    2) f(2)-f(1)=f'(c_2)(2-1)\implies f'(c_2)=0
    3) f(2)-f(1)=f'(c_3)(2-0)\implies f'(c_3)=1/2
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