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Math Help - Calculus Tangency

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    Calculus Tangency

    Let f(x) = 3-x^2, x _<2 (here x is less than or equal to 2)
    4x-9, x<2

    Determine whether the graph of y=f(x) has a tangent at x=2. If not, explain why not.
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  2. #2
    MHF Contributor red_dog's Avatar
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    f(x)=\left\{\begin{array}{ll}3-x^2, & x\geq 2\\4x-9, & x>2\end{array}\right.

    \lim_{x\nearrow 2}f(x)=\lim_{x\searrow 2}f(x)=f(2)=-1\Rightarrow f continuous in x=2.

    f is differentiable in a vicinity of x=2.

    f'(x)=\left\{\begin{array}{ll}-2x, & x<2\\4, & x>2\end{array}\right.

    f_{left}'(2)=\lim_{x\nearrow 2}f'(x)=-4

    f_{right}'(2)=\lim_{x\searrow 2}f'(x)=4

    So f is not differentiable in x=2 and the graph has not a single tangent. The graph has two semitangents (x=2 is an angular point)
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