Increasing and decreasing intervals: Class 12 practice
f is increasing where f′(x) > 0 and decreasing where f′(x) < 0. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Differentiation
- Sign of a quadratic
The idea behind increasing and decreasing intervals
f is increasing where f′(x) > 0 and decreasing where f′(x) < 0. Find where f′(x) = 0, then test the sign of f′ in each interval.
Step-by-step method
- Find f′(x).
- Solve f′(x) = 0 to get critical points.
- Test the sign of f′ in each interval.
- Write the intervals.
Worked example 1 (Basic)
Find where f(x) = x² + 2x is increasing.
- f′(x) = 2x + 2
- f′ > 0 when x > -1
- Answer: (-1, ∞)
Worked example 2 (Standard)
f′(x) = 3(x + 3)(x + 1). On which intervals is f increasing and decreasing?
- Critical points x = -3, -1
- f′ > 0 for x < -3 and x > -1; f′ < 0 between
- Answer: Increasing on (−∞, -3) ∪ (-1, ∞); decreasing on (-3, -1)
Worked example 3 (Challenge)
f′(x) = 3(x + 1)(x − 1). On which intervals is f increasing and decreasing?
- Critical points x = -1, 1
- f′ > 0 for x < -1 and x > 1; f′ < 0 between
- Answer: Increasing on (−∞, -1) ∪ (1, ∞); decreasing on (-1, 1)
Common mistakes and misconceptions
- Testing f instead of f′
- Including endpoints carelessly
Practice questions
- (Basic) Find where f(x) = x² + 8x is increasing.
- (Basic) Find where f(x) = x² − 2x is increasing.
- (Standard) f′(x) = 3(x + 3)(x − 1). On which intervals is f increasing and decreasing?
- (Standard) f′(x) = 3(x + 0)(x − 2). On which intervals is f increasing and decreasing?
- (Challenge) f′(x) = 3(x − 1)(x − 4). On which intervals is f increasing and decreasing?
- (Challenge) f′(x) = 3(x + 2)(x + 1). On which intervals is f increasing and decreasing?
Answer key with full working
- Q1: f′(x) = 2x + 8 → f′ > 0 when x > -4 → Answer: (-4, ∞)
- Q2: f′(x) = 2x − 2 → f′ > 0 when x > 1 → Answer: (1, ∞)
- Q3: Critical points x = -3, 1 → f′ > 0 for x < -3 and x > 1; f′ < 0 between → Answer: Increasing on (−∞, -3) ∪ (1, ∞); decreasing on (-3, 1)
- Q4: Critical points x = 0, 2 → f′ > 0 for x < 0 and x > 2; f′ < 0 between → Answer: Increasing on (−∞, 0) ∪ (2, ∞); decreasing on (0, 2)
- Q5: Critical points x = 1, 4 → f′ > 0 for x < 1 and x > 4; f′ < 0 between → Answer: Increasing on (−∞, 1) ∪ (4, ∞); decreasing on (1, 4)
- Q6: Critical points x = -2, -1 → f′ > 0 for x < -2 and x > -1; f′ < 0 between → Answer: Increasing on (−∞, -2) ∪ (-1, ∞); decreasing on (-2, -1)
Syllabus note
This skill is listed under "Application of Derivatives" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is increasing and decreasing intervals in?
"Application of Derivatives", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.