Is the function f(x) = x|x − 5| − 5|x − 5| differentiable at the point x = 5? Justify your answer (a) using the limit definition of the derivative at x = 5 and (b) graphing f(x) and using your graph to draw a rough sketch of f'(x) (it does not need to be to scale).

Question

Is the function f(x) = x|x − 5| − 5|x − 5| differentiable at the point x = 5? Justify your answer (a) using the limit definition of the derivative at x = 5 and (b) graphing f(x) and using your graph to draw a rough sketch of f'(x) (it does not need to be to scale).

Answer 1

f(x) = (x-5)|x-5|

if x>5 this is just (x-5)^2
if x<5 this is -(x-5)^2
both branches are differentiable, the two tangents are the same at x=5, so f(x) is differentiable.

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