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calculus

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given h(x) =kx+1, X<1
(k is constant) X^2, x > and equal to 1

A) find value of k such that h(x) is continous at x=1
B) determine if h(x) is differentiable at x=1

  • calculus -

    a)
    For h(x) to be continuous,
    we need to address the point x=1, since the rest are continuous.
    h(x)=x² for x≥1 means that
    h(1)=1²=1
    We therefore require that
    lim h(x) x->1- = 1, or
    set h(x)=kx+1 = 1 => k=0
    Therefore h(x)=0*x+1 =1

    b) h'(1-) = d(1)/dx =0
    h'(1)=h'(1+)=d(x²)/dx = 2x = 2

    Since h'(1-)≠h'(1+), we conclude that h(x) is not differentiable at x=1.

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