Question

Question found in screenshot, it's only one question however, NOT 3.

Answer 1

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

$\lim _{x\rightarrow2^-}f(x)\text{ = }a(2)^2+b(2)$$\lim _{x\rightarrow2^-}f(x)=4a+2b$$\lim _{x\rightarrow2^{+^{}}}f(x)=5(2)-10=0$

Step 3

For f to be continuous

$\lim _{x\rightarrow2^-}f(x)=\lim _{x\rightarrow2^+}f(x)_{}$

Hence,

$\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}$$\begin{gathered} \text{Divide through by -2} \\ \frac{-2b}{-2}=\frac{4a}{-2} \\ b=-2a \end{gathered}$

Hence, the relationship between a and b so that f(x) is continuous at x= 2 is seen below as;

b=-2a