1.

Determine the value ofthe constant `k`so that the function `f(x)={k x^2 , if xlt=2 3 , if x >2`is continuous.

Answer» Since a polynomial funcation is continuous and a constant function is continuous , the given function is continuous for all x lt 2and for all x gt2
So, consider the point x=2 , we have f(2) = 4k .
`lim_( xto 2+) f(x) =lim_( h to0) f( 2+h) =lim_( h to0) 3=3`
And, ` lim_( x to 2-) f(x) =lim_( h to 0) f( 2-h) =lim_( h to 0) k( 2-h)^(2) = 4k`
for continuity , we must have 4k=3 , or ` k = 3/4`


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