Let
$$f ( x ) = \left\{ \begin{array} { c c }
x - 4 & \text { if } x \leq 5 \
k x ^ { 2 } - 24 x + 46 & \text { if } x 5
\end{array} \right.$$
Find the value of $k$ that will make $f$ continuous on $( - \infty , \infty )$.
A) 5
B) 46
C) $- 4$
D) 3

Question

Let
$$f ( x ) = \left\{ \begin{array} { c c } 
x - 4 & \text { if } x \leq 5 \
k x ^ { 2 } - 24 x + 46 & \text { if } x > 5
\end{array} \right.$$
Find the value of $k$ that will make $f$ continuous on $( - \infty , \infty )$.
A) 5
B) 46
C) $- 4$
D) 3

Quizplus · Accepted Answer

For $f$ to be continuous at $x=5$, we need to have
$$\lim_{x\to5^-}f(x)=\lim_{x\to5^+}f(x)=f(5)$$
The left-hand limit is
$$\lim_{x\to5^-}f(x)=\lim_{x\to5^-}(x-4)=1$$
The right-hand limit is
\begin{align*}
\lim_{x\to5^+}f(x)&=\lim_{x\to5^+}(kx^2-24x+46)\
&=k\lim_{x\to5^+}(x-5)^2+21\
&=k\cdot0+21=21
\end{align*}
Therefore, we need to solve the equation $k\cdot0+21=1$, which gives $k=-20$. Hence, the answer is $\boxed{\text{(D)}}$.

Let f(x)={x−4 if x≤5kx2−24x+46 if x>5f ( x ) = \left\{ \begin{array} { c c } x - 4 & \text { if } x \leq 5 \\k x ^ { 2 } - 24 x + 46 & \text { if } x > 5\end{array} \right.f(x)={x−4kx2−24x+46​ if x≤5 if x>5​

Let $f ( x ) = \left\{ \begin{array} { c c } x - 4 & \text { if } x \leq 5 \\k x ^ { 2 } - 24 x + 46 & \text { if } x > 5\end{array} \right.$