Question 6 (5 points) $$ \checkmark $$ Saved For what value of the constant c is the function $$ f(\mathrm{x}) $$ continuous on the interval $$ (-\infty, \infty) $$ where $$ f(x)=\left\{\begin{array}{ll}c x^{2}+3 & \text { if } x \leq-1 \ 4 c x-6 & \text { if } x-1\end{array}\right. $$ If the answer in infinity, you can type in INF in the blank. If the answer is negative infinity, you can type -INF. If the answer is does not exist, you can type in DNE. Otherwise, write the number as an integer or reduced fraction.

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Question 6 (5 points) $$ \checkmark $$ Saved For what value of the constant c is the function $$ f(\mathrm{x}) $$ continuous on the interval $$ (-\infty, \infty) $$ where $$ f(x)=\left\{\begin{array}{ll}c x^{2}+3 & \text { if } x \leq-1 \ 4 c x-6 & \text { if } x>-1\end{array}\right. $$ If the answer in infinity, you can type in INF in the blank. If the answer is negative infinity, you can type -INF. If the answer is does not exist, you can type in DNE. Otherwise, write the number as an integer or reduced fraction.

UpStudy AI · Accepted Answer

To ensure the function $$ f(x) $$ is continuous on $$ (-\infty, \infty) $$, we need to make sure there is no discontinuity at $$ x = -1 $$, where the definition of the function changes.

Given:
$$
f(x) = \begin{cases} 
c x^{2} + 3 & \text{if } x \leq -1 \
4c x - 6 & \text{if } x > -1 
\end{cases}
$$

1. **Evaluate $$ f(x) $$ from the left at $$ x = -1 $$**:
   $$
   f(-1) = c(-1)^2 + 3 = c + 3
   $$

2. **Evaluate $$ f(x) $$ from the right at $$ x = -1 $$**:
   $$
   \lim_{x \to -1^+} f(x) = 4c(-1) - 6 = -4c - 6
   $$

3. **Set the left-hand limit equal to the right-hand limit for continuity**:
   $$
   c + 3 = -4c - 6
   $$
   $$
   5c = -9 \implies c = -\frac{9}{5}
   $$

**Final Answer:** $$-\dfrac{9}{5}$$
The value of $$ c $$ that makes the function $$ f(x) $$ continuous on $$ (-\infty, \infty) $$ is $$ -\dfrac{9}{5} $$.

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