The function f′(x) is (A) increasing in(-t, - 1/4) and decreasing in (-1/4, t)

Question

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

The function f′(x) is 

(A) increasing in(-t, - 1/4) and decreasing in (-1/4, t)

(B) decreasing in (-t, - 1/4) and increasing in (- 1/4, t)

(C) increasing in (-t, t) 

(D) decreasing in (-t, t)

Answer 1

Answer is (D) decreasing in (-t, t)

f(x) = 4x³ + 3x² + 2x + 1 

f′(x) = 12x² + 6x + 2 

f′′(x) = 2[12x + 3] = 0 

⇒ x = -1/4 

f′′′(x) = 24 

So, the funtion is decreasing in (-t, t).