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Question
Determine whether the following function is differentiable at the indicated values.
f(x) = x |x| at x = 0
Solution
`f(x) = {{:(x(- x), "if" x < 0),(x(x), "if" x > 0):}`
`f(x) = {{:(- x^2, "if" x < 0),(x^2, "if" x > 0):}`
To find the left limit of `f(x)` at x = 0
When x → 0
`f(x) = - x^2`
`f"'"(0^-) = lim_(x -> 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^-) (-x^2 - 0^2)/(x - 0)`
= `lim_(x -> 0^-) (-x^2)/x`
= `lim_(x -> 0^-) (- x)` = 0 .........(1)
To find the right limit of `f(x)` at x = 0
When x → 0+
`f(x) = x^2`
`f"'"(0^+) = lim_(x -> 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^+) (x^2 - 0^2)/x`
= `lim_(x -> 0^+) (x^2)/x`
= `lim_(x -> 0^+) x` = 0 .........(2)
From eqation (1) and (2) we get
`f"'"(0-) = f"'"(0^+)`
∴ f(x) is differentiable at x = 0.
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