If 2x3+ax2+bx+4=0  (a and b are positive real numbers) has three real roots. The minimum value of a3 isThe minimum value of b3 is The minimum value of (a + b)3 is

∵AM≥GM∴ (−α)+(−β)+(−γ)3≥{(−α)(−β)(−γ)}1/3a23≥(2)1/3a≥6(2)1/3----ior  a3≥432Hence, minimum value of a3 is 432.∵AM≥GM∴(−α)(−β)+(−β)(−γ)+(−γ)(−α)3⇒≥{(−α)(−β)(−β)(−γ)(−γ)(−α)}1/3⇒b/23≥(4)1/3----ii⇒b≥6(4)1/3or  b3≥864Hence, minimum value of b3 is 864From Eqs. (i) and (iü), we get ab≥6(2)1/3⋅6(4)1/3⇒ ab≥36×2∵ a+b2≥ab≥62⇒a+b2≥62∴ a+b≥122or (a+b)3≥34562Hence, minimum value of (a + b)3 is 34562