JEE Advanced2008MathematicsApplication of DerivativesActual
Paragraph: Consider the function f:(- , ) (- , ) defined by f(x)= x^2-a x+1 x^2+a x+1 ; 0 Question: Let g(x)= ₀^ e^x f^ (t) 1+t^2 d t . Which of the following is true ?
Options
- Ag^ (x) is positive on (- , 0) and negative on (0, )
- Bg^ (x) is negative on (- , 0) and positive on (0, )
- Cg^ (x) changes sign on both (- , 0) and (0, )
- Dg^ (x) does not change sign on (- , )
Correct answer
B. g^ (x) is negative on (- , 0) and positive on (0, )
Step-by-step solution
g^ (x)= f^ (e^x ) 1+ (e^x )^2 e^x=2 a [ e^ 2 x -1 (e^ 2 x +a e^x+1 )^2 ] ( e^x 1+e^ 2 x ) aligned & g^ (x)=0 , if e^ 2 x -1 & =0, i.e., & & x=0 If & & x < 0, e^ 2 x < 1 g^ (x) < 0 . aligned