Find constants $a \text { and } b$ so that $f ( x ) = \left\{ \begin{array} { l l }
a x + b & \text { if } x

Question

Find constants $a \text { and } b$ so that $f ( x ) = \left\{ \begin{array} { l l } 
a x + b & \text { if } x < 1 \
b x ^ { 2 } + 1 & \text { if } x \geq 1
\end{array} \right.$ , is continuous and differentiable everywhere?
A) $a = 2 , b = 1$
B) $a = 1 , b = 1$
C) $a = 1 , b = \frac { - 1 } { 2 }$
D) $a = 1 , b = \frac { 1 } { 2 }$

Quizplus · Accepted Answer

To ensure continuity at $x = 1$, $a(1) + b = b(1)^2 + 1$. To ensure differentiability, the derivatives from both sides at $x = 1$ must be equal: $a = 2b(1)$. Solving these gives $a = 1$ and $b = \frac{1}{2}$.

Find Constants a and ba \text { and } ba and b So That f(x)={ax+b if x<1bx2+1 if x≥1f ( x ) = \left\{ \begin{array} { l l } a x + b & \text { if } x < 1 \\b x ^ { 2 } + 1 & \text { if } x \geq 1\end{array} \right.f(x)={ax+bbx2+1​ if x<1 if x≥1​

Find Constants $a \text { and } b$ So That $f ( x ) = \left\{ \begin{array} { l l } a x + b & \text { if } x < 1 \\b x ^ { 2 } + 1 & \text { if } x \geq 1\end{array} \right.$