If `f(x)={(ax+3",",x le 2),(a^(2)x-1"," , x gt 2):}`, then the values of a for which f is continuous for all x are

Options

A. a and -2
B. 1 and 2
C. `-1 and 2 `
D. `-1 and -2`

Answer

C

Explanation

Given, `f(x)={(ax+3",",x le 2),(a^(2)x-1",",x gt 2):},` continuity at x = 2,
`LHL=lim_(x to 2^(-)) f(x)=lim_( x to 2) (ax+3)=2a+3`
`RHL=lim_(x to 2^(+)) f(x)=lim_(x to 2)(a^(2)x-1)=2a^(2)-1`
Since, f(x) is continuous for all values of x.
`therefore LHL=RHL`
`rArr 2a+3=2a^(2)-1`
`rArr 2a^(2)-2a-4=0`
`rArr a^(2)-a-2=0 rArr a^(2)-2a+a-2=0`
`rArr a(a-2)+1(a-2)=0 rArr (a+1)(a-2)=0`
`therefore a = -1,2`