Question

For what values of a and b is

g(x)={ax+2b, x<=0

x²+3a-b, 0<x<=2

3x-5, x>2}

continuous at every x?

Answer 1

if g(x) is continuous at every x then

$\mathop {\lim }\limits_{x \to 0 - 0} g(x) = \mathop {\lim }\limits_{x \to 0 + 0} g(x) \Rightarrow a \cdot 0 + 2b = {0^2} + 3a - b$

$\mathop {\lim }\limits_{x \to 2 - 0} g(x) = \mathop {\lim }\limits_{x \to 2 + 0} g(x) \Rightarrow {2^2} + 3a - b = 3 \cdot 2 - 5$

We have a system of equations:

$\left\{ \begin{matrix} 2b = 3a - b\\ 4 + 3a - b = 1 \end{matrix} \right.$

$a = b = - \frac{3}{2}$

Answer: $a = b = - \frac{3}{2}$