Derivatives 5

1. (a+1/a)-(b+ 1/b)
= (a-b)- (a-b)/(ab)
= (a-b)(ab-1)/(ab) < 0
so, (a+1/a) < (b+ 1/b)
2. f'(x)=3ax²+2bx+c= 3a(x+2)(x-1) (since f'(-2)=0=f'(1) )
3ax²+2bx+c= 3a(x²+x-2), so, 2b= 3a , c= -6a
f(x)= ax³ +1.5ax²-6ax+d
f(-2)=3= -8a+6a+12a+d, 10a+d=3
f(1)=0 = a+1.5a-6a+d, -3.5a+d=0
a=3/13.5= 2/9
b= 1.5a= 1/3
c= -6a= -4/3
d=7/9
f(x)=(2x³+3x²-12x+7)/9