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

Answer 1

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