Increasing & Decreasing Functions, Maxima & Minima

Increasing & Decreasing Functions

Strictly Increasing Function

• A function f(x) is strictly increasing on (a, b) if:
• x₁ < x₂ ⇒ f(x₁) < f(x₂) for all x₁, x₂ ∈ (a, b)

Strictly Decreasing Function

• A function f(x) is strictly decreasing on (a, b) if:
• x₁ < x₂ ⇒ f(x₁) > f(x₂) for all x₁, x₂ ∈ (a, b)

Monotonic Function

• A function is called monotonic on (a, b) if it is either increasing or decreasing on (a, b).

Relation with Derivatives

Necessary Condition

• If f(x) is increasing on (a, b), then:
• f′(x) > 0 for all x ∈ (a, b)
• If f(x) is decreasing on (a, b), then:
• f′(x) < 0 for all x ∈ (a, b)

Sufficient Condition (Theorem)

• If f′(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b)
• If f′(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b)

Properties of Monotonic Functions

• If f(x) is strictly increasing on [a, b], then f⁻¹(x) exists and is strictly increasing
• If f(x) is continuous and strictly increasing on [a, b], then f⁻¹(x) is continuous on [f(a), f(b)]
• If f′(x) ≥ 0 (or > 0) for all x ∈ (a, b), then f(x) is monotonically (or strictly) increasing
• If f(x) and g(x) are both increasing (or decreasing), then g∘f(x) is also increasing (or decreasing)
• If one function is increasing and the other decreasing, then g∘f(x) is decreasing

Maxima & Minima

Maximum Value

• f(x) attains maximum at x = a if:
• f(x) ≤ f(a) for all x in the domain
• a is called the point of maximum

Local Maximum

• f(x) has a local maximum at x = a if:
• f(x) < f(a) for all x in a neighbourhood of a (excluding a)

Local Minimum

• f(x) has a local minimum at x = a if:
• f(x) > f(a) for all x in a neighbourhood of a (excluding a)

Necessary Condition for Extreme Value

• If f(x) has a local maximum or minimum at x = a and f′(a) exists, then:
• f′(a) = 0
• Such points are called stationary or critical points
• Note: f′(a) = 0 is only a necessary condition, not sufficient

First Derivative Test

• Local Maximum at x = a if:
• f′(a) = 0
• f′(x) changes from positive to negative at x = a
• Local Minimum at x = a if:
• f′(a) = 0
• f′(x) changes from negative to positive at x = a
• If f′(a) = 0 and sign does not change ⇒ no extremum

Higher Order Derivative Test

• If f′(c) = f″(c) = … = f⁽ⁿ⁻¹⁾(c) = 0 and f⁽ⁿ⁾(c) ≠ 0:
• If n is even and f⁽ⁿ⁾(c) < 0 ⇒ local maximum at x = c
• If n is even and f⁽ⁿ⁾(c) > 0 ⇒ local minimum at x = c
• If n is odd ⇒ neither maximum nor minimum

Concavity & Point of Inflection

Concavity

• Curve is concave upward if it lies above its tangent
• Curve is concave downward if it lies below its tangent

Point of Inflection

• A point where curve changes concavity
• x = c is a point of inflection if:
• f″(c) = 0 or f″(c) does not exist
• and f″(x) changes sign at x = c
• Alternatively, if f‴(c) ≠ 0

Critical Points

• x = α is a critical point if:
• f′(α) = 0 or f′(α) does not exist

JEE Main Focus Tips

• Always find f′(x) first and analyse its sign
• Check derivative sign intervals carefully
• Don’t assume f′(a) = 0 implies extremum
• Absolute value functions are common traps
• Inflection point questions often involve f″(x)