Let x=-1 and x=2 be the critical points of the function f ( x )= x ^3+ ax ^2+ b _ c | x |+1, x 0 . Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [-2,- 1 2 ] . Then | M +m| is equal to (Take _ c 2=0.7 ):

Step-by-step solution

aligned & f ( x )= x ^2+ ax ^2+ b n | x |+1, x 0 & f ^ ( x )=3 x ^2+2 ax + b x & f ^ (-1)=3-2 a - b =0 & f ^ (-2)=12+4 a - b 2 =0 & a = -9 2 , ~b =12 & f ^ ( x )=3 x ^2-9 x + 12 x = 3( x +1)( x +2)^2 x aligned Max. at n =-1 f(x)=x^2- 9 2 x^2+12 |x|+1 aligned & f (-1)=-1- 9 2 +1=- 9 2 & M =-4.5 aligned Min. value at x =-2 aligned & f (-2)=-8-18+12 n 2+1 & ~m =-25+12 n 2=-16.6 & | M + m |=21.1 aligned