If `f(x)={((sin((2x^2)/a)+cos((3x)/b))^(ab//x^2),;x!=0) ,(e^3 ,; x=0):}`
is continuous at `x=0AAb in R` then minimum value of `a` is

Options

A. `-1//8`
B. `-1//4`
C. `-1//2`
D. 0

Answer

B

Explanation

`f(x)` is continuous at x = 0
`therefore" "underset(xrarr0)(lim)(sin.(2x^(2))/(a)+cos.(3x)/(b))^(ab//x^(2))=f(0)`
`rArr" "e^(3)=e^(underset(xrarr0)(lim)(sin.(2x^(2))/(a)+cos.(3x)/(b)-1)(ab)/(x^(2)))`
`rArr" "3=abunderset(xrarr0)(lim)((2)/(a)(sin(2x^(2))/(a))/((2x^(2))/(a))-(9)/(4b^(2))(2sin^(2).(3x)/(2b))/((9x^(2))/(4b^(2))))`
`rArr" "3-ab((2)/(a)-(9)/(2b^(2)))`
`rArr" "3=ab((4b^(2)-9a)/(2ab^(2)))`
`rArr" "3=(4b^(2)-9a)/(2b)`
`rArr" "4a^(2)-6b-9a=0`
Since b is real, we have
`d ge 0`
`rArr" "36+144a ge 0`
`rArr" " a ge -1//4`