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Question
Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`
Solution
If `x ≥ -3/2`, |2x + 3| = 2x + 3 and if `x < -3/2`, |2x + 3| = − (2x + 3)
∴ f(x) `{:(= (2x + 3)^2"," , "for" x ≥ - 3/2),(= -(2x + 3)^2"," , "for" x < - 3/2):}`
`"R" "f'"(-3/2) = lim_("h" -> 0) ("f"(- 3/2 + "h") - "f"(-3/2))/"h"`
= `lim_("h" -> 0) ([2(- 3/2 + "h") + 3]^2 - [2(- 3/2) + 3]^2)/"h" ...[because "f"(x) = (2x + 3)^2"," "for" x ≥ - 3/2]`
= `lim_("h" -> 0) ([(-3 + 2"h") + 3]^2 - 0)/"h"`
= `lim_("h" -> 0) (4"h"^2)/"h"`
= `lim_("h" -> 0) (4"h")` ...[∵ h → 0 ∴ h ≠ 0]
= 0
Similarly, `"L" "f'"(- 3/2)` = 0
∴ `"R""f'"(- 3/2) = "L" "f'"(- 3/2)` = 0
∴ f is differentiable at x = `- 3/2` and hence continuous at x = `-3/2`.
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