Question

Find the values of $k$ so that the function $f$ is continuous at the indicated point:
$f\left(x\right)=$ $\{\begin{array}{c}k{x}^2,\text{if}x\le \pi \\ \cos x,\text{if}x>\pi \end{array}$

at $x=$ $\pi$

Answer 1

For $f\left(x\right)$ to be continuous at $x=\pi$

$\underset{x\to {\pi }^{-}}{lim}f\left(x\right)=\underset{x\to {\pi }^{+}}{lim}f\left(x\right)$

$\Rightarrow k{\pi }^2=\cos \pi$

$\Rightarrow k{\pi }^2=-1$

$\Rightarrow k=-\frac{1}{{\pi }^2}$