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Question
Select the correct answer from the given alternative:
If f(x) `{:( = x^2 + sin x + 1, "for" x ≤ 0),(= x^2 - 2x + 1, "for" x ≤ 0):}` then
Options
f is continuous at x = 0, but not differentiable at x = 0
f is neither continuous nor differentiable at x = 0
f is not continuous at x = 0, but differentiable at x = 0
f is both continuous and differentiable at x = 0
Solution
f is continuous at x = 0, but not differentiable at x = 0
Explanation;
f(x) `{:( = x^2 + sin x + 1,"," x ≤ 0),(= x^2 - 2x + 1,"," x ≤ 0):}`
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) (x^2 + sinx + 1)` = 0 + 0 + 1 = 1
`lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) (x^2 - 2x + 1)` = 0 – 0 + 1 = 1
∴ f is continuous at x = 0
Lf'(0) = `lim_("h" -> 0^-) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^-) ("h"^2 + sin "h" + 1 - (0 + 0 + 1))/"h"`
= `lim_("h" -> 0^-) ("h" + sin"h"/"h")` = 0 + 1 = 1
Rf'(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^+) ("h"^2 - 2"h" + 1 - 1)/"h"`
= `lim_("h" -> 0) ("h" - 2)`
= – 2
∵ Rf'(0) ≠ Lf'(0)
∴ f is not differentiable at x = 0.
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