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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
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Solution
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
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