Question

Select the correct answer from the given alternative:

If f(x) `{:(= 2x + 6, "for"  0 ≤ x ≤ 2),(= "a"x^2 + "b"x, "for"  2 < x ≤4):}` is differentiable at x = 2 then the values of a and b are

Options

• a = `- 3/2`, b = 3

• a = `3/2`, b = 8

• a = `1/2`, b = 8

• a = `- 3/2`, b = 8

Answer 1

a = `- 3/2`, b = 8

Explanation;

f(x) `{:(= 2x + 6,","  0 ≤ x ≤ 2),(= "a"x^2 + "b"x,","  2 < x ≤4):}`

Lf'(2) = 2, Rf'(2) = 2a (2) + b

∵ Lf'(2) = Rf'(2) .......[f is differentiable]

∴ 2 = 4a + b …(i)

∴ f is continuous

∴ `lim_(x -> 2^+) "f"(x) = "f"(2) = lim_(x -> 2^-) "f"(x)`

∴ 4a + 2b = 2(2) + 6

∴ 4a + 2b = 10

∴ 2a + b = 5 …(ii)

Solving (i) and (ii),

a = `-3/2`, b = 8