Find the values of a and b that make f continuous everywhere.

Question

f(x) =

ax² + x - b       if x ≤ 2

ax + b         if 2 < x < 5

2x − 7           if x ≥ 5

Answer 1

By definition of continuity, we check that the limit of a point is the same when it is approached from the left side of it and from from the right side. In this case let’s examine f as x approaches 2 from the left we get

4a+2-b.

As x approaches 2 from the right

2a+5.

equating these two expressions simplify to

b = 2a -3.

we do the same for the next point 5. By reapplying the definition of continuity for x=5,

5a+b=3.

From here it is left to find the coefficient a and b.

hope this works