Let (2,3) be the largest open interval in which the function f(x)=2 _ e (x-2)-x^2+a x+1 is strictly increasing and (b, c) be the largest open interval, in which the function g (x)=(x-1)^3(x+2- a )^2 is strictly decreasing. Then 100(a+b-c) is equal to :

Step-by-step solution

aligned & f ^ ( x )= 2 x -2 -2 x + a 0 & f ^ ( x )= -2 ( x -2)^2 -2 < 0 & f ^ ( x ) & f ^ (3) 0 & 2-6+ a 0 & a 4 & a _ =4 & ~g ( x )=( x -1)^3( x +2- a )^2 & ~g ( x )=( x -1)^3( x -2)^2 & ~g ^ ( x )=( x -1)^3 2( x -2)+( x -2)^2 3( x -1)^2 & =( x -1)^2( x -2)(2 x -2+3 x -6) & =( x -1)^2( x -2)(5 x -8) < 0 & x ( 8 5 , 2 ) & 100( a + b - c )=100 (4+ 8 5 -2 )=360 aligned