There is a cubic polynomial f(x) with values of x lying in the interval [-1,2] . Given the condition. (i) f^ (x)=24 (ii) An extreme of f^ (x) lies at x= -1 6 (iii) The coefficient of x and x^0 in f(x) are 0 and 6 respectively. Find the greatest value of f(x) .

Step-by-step solution

Since the coefficient of x and x^0 are 0 and 6 respectively, let f(x)=a x^3+b x^2+6 . Then aligned f(x) & =a x^3+b x^2+6 f^ (x) & =3 a x^2+2 b x f^ (x) & =6 a x+2 b f^ (x) & =6 a=24 aligned Since f^ (x)=6 a=24 a=4 Since an extreme of f^ (x) occurs when x= 1 6 f^ ( 1 6 )=6 a ( -1 6 )+2 b=-4+2 b=0 b=2 Therefore f(x)=4 x^3+2 x^2+6 and f^ (x)=12 x^2+4 x>0 for x>0 Therefore f(x) is an increasing function for x>0 and it is maximum when x=2 (f(x))=f(2)=4 (2^3 )+2 (2^2 )+6=46