Question

The function f(x) = x3 - 5x2 + 7x + 4 is a strictly increasing function in the interval:
1.  (-∞, 1) ∪ (7/3, ∞)
2.  (-∞, -1) ∪ (7/3, ∞)
3. (1, 7/3)
4. None of these.

Answer 1

Correct Answer - Option 1 :  (-∞, 1) ∪ (7/3, ∞)

Concept:

For a function y = f(x):

• At the points of local maxima or minima, f'(x) = 0.
• In the regions where f(x) is increasing, f'(x) > 0.
• In the regions where f(x) is decreasing, f'(x) < 0.

 

Calculation:

f(x) = x3 - 5x2 + 7x + 4

⇒ f'(x) = 3x² - 10x + 7

For f(x) to be increasing, f'(x) > 0.

⇒ 3x2 - 10x + 7 > 0

⇒ 3x2 - 7x - 3x + 7 > 0

⇒ x(3x - 7) - (3x - 7) > 0

⇒ (3x - 7)(x - 1) > 0

⇒ [3x - 7 > 0 AND x - 1 > 0] OR [3x - 7 < 0 AND x - 1 < 0]

⇒ [x > 7/3 AND x > 1] OR x < 7/3 AND x < 1

⇒ x > 7/3 OR x < 1

⇒ x ∈ (7/3, ∞) ∪ (-∞, 1)