Given
F(x) = ax+3, x>5
8 , x=5
x^2 + bx + a, x<5
Find a and b so that f(x) is continuous everywhere.
For x to be continuous everywhere,
Lim x→5 =8 .....(1)
Lim x→5- =8,.....(2), and
Lim x→5+ =8,.....(3)
(1) is satisfied by setting
F(5)=8
(2) can be satisfied by setting
F(5)=ax+3 = 8
and solving for a.
(3) can be satisfied by setting
F(5) = x^2 + bx + a = 8
and solving for b using the previously solved value of a.
Can you take it from here?