Tutorial Exercise

Let $f(x)=\{\begin{array}{ccc}c{x}^2& 7xif& x<mn>4</mn>\\ x^3-cx& if& x4\end{array}$

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

Step $1$

Recall that a function $f$ is continuous at a number a if the following holds.

$\underset{xa}{lim}f(x)=f(a)$

We are given the following.

$f(x)=\{\begin{array}{ccc}c{x}^2& 7xif& x<mn>4</mn>\\ x^3-cx& if& x4\end{array}$

We note that for all values of the constant $c$ the function $f$ is continuous on both $(-,4)$ and $(4,).$ The only case that we need to consider is when $x=4.$ For the function $f$ to be continuous on $(-,),$ we need to ensure that there is not a discontinuity at $x=4.$ To do so$,$ we must determine the value of $c$ such that the following holds.

$\underset{x{4}^{-}}{lim}f(x)=\underset{x{4}^{}}{lim}f(x)$

Find the following limits$.$

$\underset{x{4}^{-}}{lim}f(x)=\underset{x{4}^{-}}{lim}(c{x}^{2}7x)$

$=$

$\underset{x{4}^{}}{lim}f(x)=\underset{x{4}^{}}{lim}(x^3-cx)$

$=$

$$

SKIP $($YOU CANNOT COME BACK$)$