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 the right-hand limit to be equal to the left-hand limit, and equal to $f(5)$. Using the fact that $f(5) = 5 - 4 = 1$, we get:
\begin{align*}
\lim_{x \to 5^-} f(x) &= \lim_{x \to 5^-} (x-4) \
&= 1 \
\lim_{x \to 5^+} f(x) &= \lim_{x \to 5^+} (kx^2 - 24x + 46) \
&= k \cdot (5)^2 - 24(5) + 46 \
&= 25k - 34 
\end{align*}
For the limits to be equal, we need:
$$1 = 25k - 34 \Rightarrow k = \frac{35}{25} = \frac{7}{5} = 1.4$$
Therefore, the correct answer is 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 $f ( x ) = \left\{ \begin{array} { c c } x - 4 & \text { if } x \leq 5 \\k x ^ { 2 } - 24 x + 46 & \text { if } x > 5\end{array} \right.$