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For what value of the constant $ c $ is the function $ f $ continuous on $ (-\infty, \infty) $?

$ f(x) = \left\{ \begin{array}{ll} cx^2 + 2x & \mbox{if $ x < 2 $}\\ x^3 - cx & \mbox{if $ x \ge 2 $} \end{array} \right.$

$f$ is continuous on $(-\infty, 2)$ and $(2, \infty) .$ Now $\lim _{x \rightarrow 2-} f(x)=\lim _{x \rightarrow 2-}\left(c x^{2}+2 x\right)=4 c+4$ and$\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{+}}\left(x^{3}-c x\right)=8-2 c .$ So $f$ is continuous $\Leftrightarrow 4 c+4=8-2 c \Leftrightarrow 6 c=4 \Leftrightarrow c=\frac{2}{3} .$ Thus, for $f$to be continuous on $(-\infty, \infty), c=\frac{2}{3}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

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This is Problem forty five of the Stuart Calculus eighth edition section two point five For what value of the constant C is the function f continuous on negative infinity to infinity. So the interval of all real numbers function after its equal to si X squared was two x if exit, listen to and X cubed minus six. If X is greater than or equal to two separately and individually. Each of these functions are polynomial functions and so they're continuous on their domain. So continuity is confirmed for all values less than two and all values grow them too. However, in order to establish a continuity at every point, we need to establish a continuity where the functions change from the first, the second and that's had X equals to two. So we need to evaluate the limit as expert is two from the left of the function of and make sure it is equal to the limit as expert, just two from the right For this left limit, we are using the first function since it is in the domain of Ex Listen to. So let's see X squared, plus two acts equal to limit as experts just two from the right of X cubed minus C X and using our limit loss, we should get C times two squared, which is for plus two times two is also four equals to this limit. Utilizes Oh, with our limit, laws were allowed to plug in this value that X approaches into the function for X. So you get to Cube, which is eight. Come on minus C times two. Now combining like terms, we get sixty on one side this's foresee plus two seam and then he really mints for which is for and then dividing. We had four over six, which reduced is two or three. And this value guarantees that as long as he is equal to two thirds, the limits will be equal to each other. That limit exists and that limit is also equal to the function evaluated at X equals two, which is our definition of continuity on this. We have established continuity for this. He's for his function on the domain of all real numbers

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