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Thread: two functions ( common tangent in a common point )

  1. #1
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    two functions ( common tangent in a common point )

    The functions from the picture have a common tangent in a common point if:
    A. a=1+e
    B. a=0
    C. a=1
    D. a=e-pi
    E. a=-1

    I know that the conditions are: f(x)=g(x) and f'(x)=g'(x)
    I tried to solve the system but I did't get too far.
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  2. #2
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    Re: two functions ( common tangent in a common point )

    Quote Originally Posted by Vali View Post
    The functions from the picture have a common tangent in a common point if:
    A. a=1+e
    B. a=0
    C. a=1
    D. a=e-pi
    E. a=-1
    I know that the conditions are: f(x)=g(x) and f'(x)=g'(x)
    I tried to solve the system but I did't get too far.
    The functions are: $f(x)=x+\sqrt{x^2+a},~a\ge 0~\&~g(x)=x^2+1$ then $f'(x)=1+\dfrac{x}{\sqrt{x^2+a}}~\&~g'(x)=2x$

    I submit to you that there is a valid answer in that list. Look at the graph
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  3. #3
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    Re: two functions ( common tangent in a common point )

    The algebra involved is pretty much undoable by hand.

    What I suggest is that for each of those values of $a$ listed you plot $f$ and $g$ and see which value makes the two intersect at a tangent point.
    DESMOS, or some other software will help with this.

    Once you can visualize the situation the correct answer is pretty simple to see.
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