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प्रश्न
Determine whether the following function is differentiable at the indicated values.
f(x) = |x| + |x – 1| at x = 0, 1
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उत्तर
To find the limit at x = 0
First we find the left limit of f(x) at x = 0
When x = 0– |x| = – x and
|x – 1| = – (x – 1)
∴ When x = 0 we have
f(x) = – x – (x – 1)
f(x) = – x – x + 1 = – 2x + 1
f(0) = 2 × 0 + 1 = 1
`f"'"(0^-) = lim_(x -> 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^-) (-2x + 1 - 1)/x`
= `lim_(x -> 0^-) (- 2x)/x`
f'(0– = – 2 ……… (1)
∴ When x = 0+ we have
|x| = x and |x – 1| = – (x – 1)
∴ f(x) = x – (x – 1)
f(x) = x – x + 1
f(x) = 1
f(0) = 1
`f"'"(0^+) = lim_(x -> 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^+) (1 - 1)/x`
`f"'"(0^+)` = 0 .......(2)
From equations (1) and (2) , we get
f'(0–) ≠ f’(0+)
∴ f(x) is not differentiable at x = 0.
To find the limit at x = 1
First we find the left limit of f(x) at x = 1
When x = 1, |x| = x and
|x – 1| = – (x – 1)
∴ f(x) = x – (x – 1)
f(x) = x – x + 1 = 1
f(x) = 1
f(1) = 1
`f"'"(1^-) = lim_(x -> 1^-) (f(x) - f(1))/(x - 1)`
= `limm_(x -> 1^-) (1 - 1)/(x - 1)`
`f"'"(1^-) = lim_(x -> 0^-) 0/(x - 1)` = 0 ........(3)
When x = 1+ , |x| = x and
|x – 1| = x – 1
When x = 1, |x| = x and
|x – 1| = x – 1
∴ f(x) = x + x – 1 = 2x – 1
f(1) = 2 × 1 – 1 = 2 – 1 = 1
`f"'"(1^+) = lim_(x -> 1^+) (f(x) - f(1))/(x - 1)`
= `lim_(x -> 1^+) (2x - 1 - 1)/(x - 1)`
= `lim_(x -> 1^+) (2x - 2)/(x - 1)`
= `lim_(x -> 1^+) (2(x - 1))/(x - 1)`
`f"'"(1^-) = lim_(x -> 1^-) (2)` = 2 .........(4)
From equations (3) and (4), we get
f’(1–) ≠ f'(1+)
∴ f(x) is not differentiable at x = 1
