Find the values of a and b such that the function is continuous on interval (-∞,∞).

Question

The function is a piecewise function:

acosx + 2 if x < 0

{

be^3x +ax² if x ≥0

Answer 1

Hello!

To solve this we need to make sure that our piecewise function is continuous at every point, which means it has a defined value at every point. Luckily, it looks like we only have to consider one point! Both of the functions in the piecewise equation are continuous, so we just need to look at the point where the connect (i.e at x = 0). To make sure these are continuous, both of the functions need to have the same value at x = 0, meaning that we can plug in 0 for x and set the equations equal to each other:

acos(0) + 2 = be³⁽⁰⁾ + a(0)²

cos(0) = 1

e⁰ = 1

a + 2 = b

This means that we can choose any values for a and b as long as b is 2 more than a.