Question

If f(x) = `{(8-6x;   0<x≤2), (4x-12;    2<x≤3),(2x+10;    3<x≤6):}` then f(x) is ______ 

Options

• continuous at x = 2 and discontinuous at x = 3 

• continuous at x = 3 and discontinuous at x = 2

• continuous at x = 2 and x = 3

• discontinuous at x = 2 and x = 3

Answer 1

`{(8-6x;   0<x≤2), (4x-12;    2<x≤3),(2x+10;    3<x≤6):}` then f(x) is continuous at x = 2 and discontinuous at x = 3.

Explanation:

`lim_{x→2^-} f(x) = lim_{x→2^-} (8 - 6x) = 8 - 12 = -4`

`lim_{x→2^+} f(x) = lim_{x→2^+} (4x - 12) = 8 - 12 = -4`

f(2) = 8 - 6(2) = -4

`lim_{x→3^-} f(x) = lim_{x→3^-}(4x - 12) = 12 - 12 = 0`

`lim_{x→3^+} f(x) = lim_{x→3^+} (2x + 10) = 6 + 10 = 16`

∴ `lim_{x→3^-} f(x) ≠ lim_{x→3^+} f(x)`

∴ f(x) is continuous at x = 2 and discontinuous at x = 3.