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Math Help - Test the differentiability of the function

  1. #1
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    Test the differentiability of the function

    Question : Test the differentiability of the function

    f(x) = \begin{cases}x^2, & x \le 1 \\ \sqrt{x}, & x > 1 \end{cases}

    at x =1
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  2. #2
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    Hello, zorro!

    Test the differentiability of the function: . f(x) \:=\: \begin{cases}x^2, & x \le 1 \\ \sqrt{x}, & x > 1 \end{cases} . at x =1

    On the left branch: . f(1) \:=\:1^2 \:=\:1

    On the right branch: . f(1) \:=\:\sqrt{1}\:=\:1

    The function is continuous at x = 1.


    On the left branch: . f'(x) \:=\:2x\quad\Rightarrow\quad f'(1) \:=\:2
    On the right branch: . f'(x) \:=\:\frac{1}{2\sqrt{x}} \quad\Rightarrow\quad f'(1) \:=\:\tfrac{1}{2}

    The function is not differentiable at x=1

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