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Question
Show that the following functions are not differentiable at the indicated value of x.
`f(x) = {{:(3x",", x < 0),(-4x",", x ≥ 0):}` , x = 0
Sum
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Solution
First we find the left limit of f(x) at x = 0
When x = 0–, then x < 0
∴ f(x) = 3x
f(0) = 3 × 0 = 0
`f"'"(0^-) = lim_(x -> 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^-) (3x - 0)/x`
= `lim_(x -> 0^-) (3)` = 3 .........(1)
Next we find the right limit of f(x) at x = 0
When x = 0+, then x ≥ 0
∴ f(x) = – 4x
f(0) = – 4 × 0 = 0
`f"'"(0^+) = lim_(x -> 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^+) (- 4x - 0)/x`
= `lim_(x -> 0^+) (-4x)/x`
= `lim_(x -> 0^+) (- 4)` = – 4 .........(2)
From equations (1) and (2), we get
f'(0–) ≠ f'((0+)
∴ f(x) is not differentiable at x = 0
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