1. ## Advice on how to solve a continous function

I have the following homework problem that I do not know how to approach.

Find c such that function defined piecewise is continuous

For what value of the constant c is the function f continuous on ( -∞, ∞)

f(x) = cx^2 + 2x if x<2
x^3 - cx if x ≥ 2

In the back of the book the answer is suppose to be 2/3. I would like advice on how to approach and solve this problem. Thanks

2. $\displaystyle \lim_{x\to 2^{-}}cx^{2}+2x=4c+4$

$\displaystyle \lim_{x\to 2^{+}}x^{3}-cx=8-2c$

$\displaystyle 4c+4=8-2c$

Solve for c and what do you get?.

3. thanks!

4. Originally Posted by fishguts
I have the following homework problem that I do not know how to approach.

Find c such that function defined piecewise is continuous

For what value of the constant c is the function f continuous on ( -∞, ∞)

f(x) = cx^2 + 2x if x<2
x^3 - cx if x ≥ 2

In the back of the book the answer is suppose to be 2/3. I would like advice on how to approach and solve this problem. Thanks
The function is defined as follows:
$\displaystyle f(x):=\begin{cases}cx^{2}+2x,&x<2 \\ cx,&x\geq2\end{cases}$
for $\displaystyle x\in\mathbb{R}$.
We know that polynomials are continuous functions and thus we see that $\displaystyle f$ is piecewise continuous; that is, $\displaystyle f$ consists of two continuous functions.
Therefore it suffices to set $\displaystyle c$ in such a way that $\displaystyle f$ is also continuous at the critical point $\displaystyle x=2$.
The rest is what galactus has shown.