Suppose f '' ( x ) exists for all real x . If f ( 2 ) = 2 , f ( 3 ) = 5 and f ( 4 ) = 10 , then which one among the following statements is definitely true?

Step-by-step solution

Consider f ( x ) = a x 2 + b x + c f ( 2 ) = 4 a + 2 b + c = 2   … I f ( 3 ) = 9 a + 3 b + c = 5   … II f ( 4 ) = 16 a + 4 b + c = 10   … III Subtract equation I from equation II , 5 a + b = 3   … IV Subtract equation II from equation III , 7 a + b = 5   … V Subtract equation IV from equation V , 2 a = 2 a = 1 Substitute the value in equation IV , 5 × 1 + b = 3 b = - 2 Now, f ' ( x ) = 2 x - 2 f '' ( x ) = 2 Since, f '' ( x ) > 1 fo