Calculus Limits: piecewise function

[Ttemme]

\(\displaystyle \mbox{11. Consider the following function:}\)

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}{lr}k^2x^3\, -\, k&x\, \leq\, 2\\3x\, -\, k^2&x\, >\, 2\end{cases}\)

\(\displaystyle \mbox{For what values of }\, k\, \mbox{ is }\, f(x)\, \mbox{continuous for all real numbers?}\)

Anyone know how to do this problem? My math teacher gave it as a "challenge" problem, so it is not required homework. I would just like to know how to complete it. Thanks!!

 

[JeffM]

\(\displaystyle \mbox{11. Consider the following function:}\)

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}{lr}k^2x^3\, -\, k&x\, \leq\, 2\\3x\, -\, k^2&x\, >\, 2\end{cases}\)

\(\displaystyle \mbox{For what values of }\, k\, \mbox{ is }\, f(x)\, \mbox{continuous for all real numbers?}\)

Anyone know how to do this problem? My math teacher gave it as a "challenge" problem, so it is not required homework. I would just like to know how to complete it. Thanks!!

For what values of k is the function \(\displaystyle g(x) = k^2x^3 - k\ continuous\ if\ x < - 2?\)

For what values of k is the function \(\displaystyle h(x) = 3x - k^2\ continuous\ if\ x > - 2?\)

So if f(x) is NOT continuous, at what value of x does that occur?

\(\displaystyle m(x)\ is\ continuous\ at\ n \iff (1)\ m(n)\ is\ real,\ and\ \displaystyle \lim_{x \rightarrow n^+}m(x) = m(n) = \lim_{x \rightarrow n^-}m(x).\)

What do you conclude from that?

 

[Ttemme]

Do you plug in the values next to the functions in for the variable X and solve for K?

 

[JeffM]

Do you plug in the values next to the functions in for the variable X and solve for K? Is this sentence even written in English? It makes no sense.

Why don't you try answering the hints I already gave you.