ohk.....to find the interval of increasing or decreasing functions, u first need to differentiate the given function wrt x.
1. d/dx(2x3-9x2-24x+7)=6x2-18x-24
now we have to find the intervals of x for which 6x2-18x-24>0 and also for which 6x2-18x-24<0.
6x2-18x-24=6(x-4)(x+1)
now when x<-1, x-4 is negative, x+1 is negative, expression positive
when -1<x<4, x-4 is negative, x+1 is positive, expression negative
when x>4, x-4 is positive, x+1 is positive, expression positive
so this expression is increasing over the intervals x<=-1 and x>=4
and decreasing over -1<=x<=4
2. d/dx(x4-2x2)=4x3-4x=4x(x+1)(x-1)
when x<-1, 4x is negative, x+1 is negative, x-1 is negative, expression negative
when -1<x<0, 4x is negative, x+1 is positive, x-1 is negative, expression positive
when 0<x<1, 4x is positive, x+1 is positive, x-1 is negative, expression negative
when x>1, 4x is positive, x+1 is positive, x-1 is positive, expression positive
so this expression is decreasing over the intervals x<=-1 and 0<=x<=1
and increasing over the intervals -1<=x<=0 and x>=1
3. d/dx(3x+x3)=3+3x2=3(x2+1)
since x2+1 is positive for all values of x, this expression is increasing for all values of x (negative infinity<=x<=positive infinity)
sory it took so long.....
