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Question on intersection of tangent and chord

  1. Sep 1, 2015 #1


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    1. The problem statement, all variables and given/known data
    Show that The tangent at (c,ec) on the curve y=ex intersects the chord joining the points (c-1,ec-1) and (c+1,ec+1) at the left of x=c

    2. Relevant equations
    Legrange's mean value theorem

    3. The attempt at a solution
    Applying LMVT at c-1, c+1
    $$f'(a)=\frac{e^c(e-\frac{1}{e})}{2}\ge f'(c)$$
    Hence the chord is parallel to the tangent at ##a## and for ex, if f'(a)>f'(c) then a>c.
    So chord has a slope greater than the slope of tangent at c. Hence it intersects at left of x=c.
    Is this correct? Are there any other methods?
  2. jcsd
  3. Sep 1, 2015 #2


    Staff: Mentor

    That's Lagrange.
    What is a? You haven't said what it is.
    You could find the equation of the tangent line at (c, ec) and find the equation of the chord through the two other points, and show that the intersection of the tangent line and chord are at a value of x less than c. That's the approach I would take, but I haven't gone all the way through to see if it is fruitful.
  4. Sep 1, 2015 #3


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    2017 Award

    I think this is what the composer of the exercise meant you to do. Don't see any other inroad.
  5. Sep 1, 2015 #4


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    Some x value between c-1 and c+1which satisfies mvt.
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