Question

Let f(x) = -55x (x<-5), 2x³ - 3x² - 120x (-5 ≤ x ≤ 4), 2x³ - 3x² - 36x - 336 (x>4). Let A = {x ∈ ℝ : f is increasing}. Then A is equal to :

• A. (-∞, -5) ∪ (4, ∞)
• B. (-5, -4) ∪ (4, ∞)
• C. (-∞, -5) ∪ (-4, ∞)
• D. (-5, ∞)

Hint:

A function is increasing where $f'(x)>0$. For piecewise functions, check the derivative in each interval separately and verify continuity at boundaries if needed.

Answer 1

The Correct Option is B

Step 1: Check derivatives. For $x<-5, f'(x) = -55<0$ (Decreasing).
Step 2: For $-5<x<4, f'(x) = 6x^2 - 6x - 120 = 6(x^2 - x - 20) = 6(x-5)(x+4)$. In $(-5, 4)$, $f'(x)>0$ when $x \in (-5, -4)$ and $f'(x)<0$ when $x \in (-4, 4)$.
Step 3: For $x>4, f'(x) = 6x^2 - 6x - 36 = 6(x^2 - x - 6) = 6(x-3)(x+2)$. For $x>4$, both $(x-3)$ and $(x+2)$ are positive, so $f'(x)>0$ (Increasing).
Step 4: $A = (-5, -4) \cup (4, \infty)$.