JEE Advanced2012MathematicsApplication of DerivativesActual
Let f(x)=(1-x)² ² x+x² for all x I R and let g(x)= ₁^ x ( 2(t-1) t+1 - t ) f(t) d t for all x (1, ) . Question: Which of the following is true?
Options
- Ag is increasing on (1, )
- Bg is decreasing on (1, )
- Cg is increasing on (1,2) and decreasing on (2, )
- Dg is decreasing on (1,2) and increasing on (2, )
Correct answer
B. g is decreasing on (1, )
Step-by-step solution
g(x)= ₁^ x ( 2(t-1) t+1 - t ) f(t) d t , g^ (x)= [ 2(x-1) x+1 - x ] f(x) Here f(x)>0, x (1, ) Let h(x)= 2(x-1) x+1 - x h^ (x)= 4 (x+1)² - 1 x = -(x-1)² (x+1)² x 1, h(x) 1 g^ (x) < 0 x (1, ) Therefore, g(x) is decreasing on (1, ) .