Question

Consider the function f(x) = 1 + 2x + 3x² + 4x³. Let s be the sum of all distinct real roots of f(x) and let t = |s|.

(i) The real number s lies in the interval

(a) ( –1/4 , 0)

(b) ( –1, – 3/4)

(c) ( – 3/4 , – 1/2)

(d) ( 0, 1/4) 

(ii) The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t lies in the interval

(a) (3/4 , 3)

(b) (21/64 , 11/16)

(c) (9, 10) (d) ( 0, 21/64)

(iii) The function f'(x) is

(a) inc. in ( – t, 1/4) and dec. in ( – 1/4 , t)

(b) dec. in ( –t, – 1/4) and inc. in ( – 1/4 , t)

(c) inc. in (– t, t)

(d) dec. in (– t, t)

Answer 1

Correct option  (i). (c), (ii). (a), (iii). (b). 

Explanation:

From the sign scheme of f"(x),

f'(x) dec. in ( –t, – 1/4) and inc. in ( –1/4 , t)