If f(x) is cubic polynomial which has local maximum at x=-1 . If f(2)=18, f(1)=-1 and f^ (x) has local minimum at x=0 , then

Step-by-step solution

Since f(x) has local maxima at x=-1 and f^ (x) has local minima at x=0 . aligned f (x) & = x f^ (x) & = x^2 2 +c 2 +c & =0 =-2 c aligned [f^ (-1)=0 ] Again, On integrating both sides, we get aligned & f(x)= x^3 6 +c x+d & f(2)= ( 8 6 )+2 c+d=18 & f(1)= 6 +c+d=-1 aligned and Using Eqs. (i), (ii) and (iii), we get aligned f(x) & = 1 4 (19 x^3-57 x+34 ) f^ (x) & = 1 4 (57 x^2-57 ) & = 57 4 (x-1)(x+1), using number line rule aligned f(x) is increasing for [1,2 5 ] and f(x) is increasing for [1,2 5 ] and f(x) has local