Question

Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = `- 3/2`

Answer 1

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`.