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Math Help - The tangent to a graph

  1. #1
    No one in Particular VonNemo19's Avatar
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    The tangent to a graph

    Here's one for ya.

    Show that if f is differentiable on an open interval I, and if the graph of f is concave upward on I, then the graph of f lies above all of its tangent lines on I.

    This is one of the problems given in my text. Although it is not required that I answer it, I feel that this problem could deepen my understanding of the second derivative test. Intuitively, I can visualize this, but I'm having trouble putting it into the language of math. Anybody got any Ideas?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Let a be any number in I

    Then the tangent line at x = a is y(x) = f ' (a)(x-a) + f(a)


    Let h(x) = f(x) - y(x) x > a

    h '(x) = f '(x) - f ' (a) > 0 for x > a since if f '' (x) is positive then f ' (x) is increasing

    Therefore h(x) is an increasing function

    h(a) = f(a) - y(a) = 0

    Therefore h(x) > h(a) > 0 i.e f(x) > y(x) for all x > a

    Since this is true for any a in I the result follows
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  3. #3
    Super Member malaygoel's Avatar
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    Quote Originally Posted by VonNemo19 View Post
    Here's one for ya.

    Show that if f is differentiable on an open interval I, and if the graph of f is concave upward on I, then the graph of f lies above all of its tangent lines on I.
    let g(x) be the equation of tangent at (x_0,y_0)

    then,
    g(x)=f'(x_0)(x-x_0)+f(x_0)

    f(x)-g(x)=f(x)-f'(x_0)(x-x_0)-f(x_0)

    Let f(x)-g(x)=h(x)

    h(x_0)=0

    h'(x)=f'(x)-f'(x_0)

    Can you interpret now?
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  4. #4
    No one in Particular VonNemo19's Avatar
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    I can follow all the way up until this point. What I'm expecting to see here is an inequality stating how every f'(x_0)<f(x_0). Am I wrong?


    Quote Originally Posted by malaygoel View Post

    h'(x)=f'(x)-f'(x_0)

    Can you interpret now?
    Last edited by VonNemo19; July 2nd 2009 at 07:15 PM.
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  5. #5
    MHF Contributor Calculus26's Avatar
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    h(x) is an increasing function therefore f(x) > g(x) for every x in I

    That is the graph of f lies above the graph of the tangent line at x0 for every x0 in I
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  6. #6
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Calculus26 View Post
    h(x) is an increasing function therefore f(x) > g(x) for every x in I

    That is the graph of f lies above the graph of the tangent line at x0 for every x0 in I
    I've got it.
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