Show that the function f (x) := x1/3, x ∈ R, is not differentiable at x = 0
Question
Show that the function f (x) := x1/3, x ∈ R, is not differentiable at x = 0
Solution
To show that the function f(x) = x^(1/3) is not differentiable at x = 0, we need to check the limit of the difference quotient as x approaches 0 from the left and from the right.
The difference quotient is given by:
(f(x) - f(0)) / (x - 0) = (x^(1/3) - 0^(1/3)) / x = x^(-2/3)
As x approaches 0 from the right (x -> 0+), the difference quotient becomes:
lim (x -> 0+) x^(-2/3) = +∞
As x approaches 0 from the left (x -> 0-), the difference quotient becomes:
lim (x -> 0-) x^(-2/3) = -∞
Since the two one-sided limits are not equal, the limit of the difference quotient as x approaches 0 does not exist. Therefore, the function f(x) = x^(1/3) is not differentiable at x = 0.
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