Question

A function is defined on R such that f(x) = (C^2)x when x≤1 and 5Cx-6 when x>1 . Determine the values of C so that f becomes continues on R

Answer 1

If f(x) is continuous at x=1 then

$\displaystyle\lim_{x\to1}f(x)=f(1)$; $\displaystyle\lim_{x\to1^-}f(x)=f(1)$; $\displaystyle\lim_{x\to1^+}f(x)=f(1)$.

$f(x) = \begin{cases} C^2x &\text{if } x\le1 \\ 5Cx-6 &\text{if } x>1 \end{cases}$

$\displaystyle\lim_{x\to1^-}f(x)=\displaystyle\lim_{x\to1}C^2x=C^2$

$\displaystyle\lim_{x\to1^+}f(x)=\displaystyle\lim_{x\to1}(5Cx-6)=5C-6$

To make $f(x)$ continuous at $x=1$ we should find $C$ for which

$C^2=5C-6$

$C^2-5C+6=0$

$C=2$ ; $C=3$ .

Answer: $C=2$ or $C=3$.