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MathsMediumClass 12

Monotonicity — Increasing/decreasing functions

Applications of Derivatives

15

JEE Qs

8%

Hard

75

min

Master the sign analysis of f'(x) using the wavy curve method and pay close attention to the function's domain and the type of monotonicity (strict vs. non-strict) required in the question.

🧮 Key Formulas

A function f(x) is strictly increasing on an interval (a,b) if for any x1, x2 in (a,b) such that x1 < x2, then f(x1) < f(x2).
A function f(x) is strictly decreasing on an interval (a,b) if for any x1, x2 in (a,b) such that x1 < x2, then f(x1) > f(x2).
If f(x) is differentiable on (a,b) and f'(x) > 0 for all x in (a,b), then f(x) is strictly increasing on (a,b).
If f(x) is differentiable on (a,b) and f'(x) < 0 for all x in (a,b), then f(x) is strictly decreasing on (a,b).
If f(x) is differentiable on (a,b) and f'(x) >= 0 for all x in (a,b), then f(x) is non-decreasing on (a,b).
If f(x) is differentiable on (a,b) and f'(x) <= 0 for all x in (a,b), then f(x) is non-increasing on (a,b).

✅ Key Points for JEE

  • 1The sign of the first derivative, f'(x), directly determines the monotonicity: f'(x) > 0 implies increasing, and f'(x) < 0 implies decreasing.
  • 2For strictly increasing/decreasing functions, f'(x) must be strictly positive/negative throughout the interval, except possibly at isolated points where f'(x) = 0.
  • 3Critical points (where f'(x) = 0 or f'(x) is undefined) are crucial; they divide the domain into sub-intervals where the sign of f'(x) can be analyzed.
  • 4Always determine the domain of the function first, as monotonicity can only be discussed within the function's valid domain.
  • 5Use proper notation for intervals (open for strict monotonicity, closed for non-decreasing/non-increasing if function is continuous at endpoints).

⚠️ Common Mistakes

  • ✕Confusing the conditions for strictly increasing (f'(x) > 0) with non-decreasing (f'(x) >= 0), especially in problems asking for 'increasing' which usually implies strict.
  • ✕Failing to identify all critical points, particularly those where f'(x) is undefined.
  • ✕Errors in solving the inequalities f'(x) > 0 or f'(x) < 0, especially for rational functions or expressions with roots of varying multiplicities.
  • ✕Not considering the original domain of the function while combining intervals or presenting the final answer.

📝 Practice Questions

See all

Q10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2

2025·MCQHard

Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________

2025·NumericalHard

Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3

2025·MCQHard

Q23.If the set of all values of a, for which the equation 5x3 −15x −a = 0 has three distinct real roots, is the interval (α, β), then β −2α is equal to ______

2025·NumericalMedium

Q13.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 1 cm/min. The surface area (in cm2 ) of the chocolate ball (without the ice- 4π cream layer) is : (1) 196π (2) 256π (3) 225π (4) 128π

2025·MCQMedium

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

2025·MCQHard

NCERT Chapters

  • Class 12 Mathematics Ch 6: Applications of Derivatives