Step 1
Given the function f defined in the question
Required: To find a relationship between a and b so that f is continuous at x=2
Step 2
[tex]\lim _{x\rightarrow2^-}f(x)\text{ = }a(2)^2+b(2)[/tex][tex]\lim _{x\rightarrow2^-}f(x)=4a+2b[/tex][tex]\lim _{x\rightarrow2^{+^{}}}f(x)=5(2)-10=0[/tex]
Step 3
For f to be continuous
[tex]\lim _{x\rightarrow2^-}f(x)=\lim _{x\rightarrow2^+}f(x)_{}[/tex]
Hence,
[tex]\begin{gathered} 4a+2b=0 \\ \text{Subtract 2b from both sides} \\ 4a+2b-2b=0-2b \\ \text{simplify} \\ 4a=-2b \\ or \\ -2b=4a \end{gathered}[/tex][tex]\begin{gathered} \text{Divide through by -2} \\ \frac{-2b}{-2}=\frac{4a}{-2} \\ b=-2a \end{gathered}[/tex]
Hence, the relationship between a and b so that f(x) is continuous at x= 2 is seen below as;
b=-2a
