Question

For what value of the constant c is the function fcontinuous on (−[infinity], [infinity])?

Answer 1

Answer:

For $c=\frac{1}{7}$ the function f(x) is continuous on $(-\infty,\infty)$.

Step-by-step explanation:

We have the following function

$f(x) = \left\{ \begin{array}{ll} cx^2+5x & \quad x$

For the function f(x) to be continuous on $(-\infty,\infty)$ it is sufficient to have continuity at x = 6, we need to ensure that as x approaches 6, the left and right limits match, this means that

$\lim_{x \to 6^{-} } f(x)=\lim_{x \to 6^{+} } f(x)=f(x)$,

which holds if and only if

$c\left(6\right)^2+5\left(6\right)=\left(6\right)^2-c\left(6\right)\\36c+30=36-6c\\42c=6$

namely if $c=\frac{1}{7}$.