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Question
If the function
f(x) = x2 + ax + b, x < 2
= 3x + 2, 2≤ x ≤ 4
= 2ax + 5b, 4 < x
is continuous at x = 2 and x = 4, then find the values of a and b
Solution
Thie function is continuous at x = 2
`lim_(x ->2^-) f(x) = lim_(x ->2^+) f(x) = f(2)`
`lim_(x ->2) x^2 + ax + b = lim_(x ->2) 3x + 2`
4 + 2a + b = 6 + 2
2a + b = 4......(1)
Given function is also continuous at x = 4.
`lim_(x ->4^-) f(x) = lim_(x ->4^+) f(x) = f(4)`
`lim_(x ->4) 3x + 2 = lim_(x ->2) 2ax + 5b`
3(4) + 2 = 2a(4) + 5b
14 = 8a + 5b........(ii)
Multiply equaiton (i) by S and subtract it from equation (ii), we get
8a + 5b = 14
10a + 5b = 20
- - -
_________________________
-2a = -6 ⇒ a = 3
Put this value of 'a' in equation (i), we get
2(3) + b = 4
b = 4 -6 = -2
Hence, a = 3 and b = -2
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