let f(x)=ax3+bx2+cx+d ∵ and (1)=−10 and f(−1)=6 ∴a+b+c+d=−10 and −a+b−c+d=6 ∵f(x) has a local minima at x=1 ∴f′(1)=0 and f′(x) has a local minima at x=−1 ∴f′′(−1)=0 ∵f(x)=ax3+bx2+cx+d ∴f′(x)=3ax2+2bx+c f′′(x)=6ax+2b ∵f′′(−1)=0 =−6a+2b=0 =b=3a...(3) also f′(1)=0 =3a+2b+c=0 =c=−9a. By adding (1) and (2), we get 2b+2d=−4 =b+d=−2 =3a+d=−2 =d=−2−3a. Put b=3a,c=−9a and d=−2−3a in (1) we get a+3a−9a−2−3a=−10 =−8a=−10+2=−8 a=
−8
−8
=1 ∴b=3,c=−9 and d=−2−3=−5 ∴f(x)=x3+3x2−9x−5 ∴f(3)=33+3.32−9×3−5 =27+27−27−5 =27−5 =22