Increasing & Decreasing Functions — Complete Formula

Basic Definition

Concept	Formula	Symbols Meaning	Key Notes / Tricks
Increasing Function	[\dfrac{dy}{dx} > 0]	[y] = function value, [x] = variable	Function rises as [x] increases ⭐
Decreasing Function	[\dfrac{dy}{dx} < 0]	—	Function falls as [x] increases ⭐

💡 Memory Hint:
Sign of derivative → behavior of function

Concept	Formula	Symbols Meaning	Key Notes
Strictly Increasing	[f'(x) > 0 ; \forall x \in I]	[I] = interval	Always rising ⭐
Strictly Decreasing	[f'(x) < 0 ; \forall x \in I]	—	Always falling ⭐

Concept	Formula	Symbols Meaning	Key Notes
Non-Decreasing	[f'(x) \ge 0]	—	Flat + increasing
Non-Increasing	[f'(x) \le 0]	—	Flat + decreasing

Concept	Formula	Symbols Meaning	Key Notes
Critical Point	[f'(x) = 0 \text{ or undefined}]	—	Possible extrema ⭐

💡 Memory Hint:
Where slope = 0 → check behavior change

Concept	Condition	Symbols Meaning	Key Notes
Increasing → Decreasing	Sign changes [+ → −]	—	Local maximum ⭐
Decreasing → Increasing	Sign changes [− → +]	—	Local minimum ⭐

💡 Memory Hint:
Sign change tells turning point

Concept	Formula	Symbols Meaning	Key Notes
Concave Up	[f”(x) > 0]	[f”(x)] = second derivative	Increasing slope ⭐
Concave Down	[f”(x) < 0]	—	Decreasing slope

💡 Memory Hint:
Second derivative → curvature

Concept	Method	Symbols Meaning	Key Notes
Interval Testing	Divide domain using critical points	—	Check sign in each interval ⭐

💡 Memory Hint:
Break → test → conclude

Concept	Use	Symbols Meaning	Key Notes
Finding Intervals	Solve [f'(x) > 0] or [< 0]	—	Direct questions ⭐
Max/Min Problems	Use derivative sign change	—	Optimization ⭐
Curve Sketching	Identify rise/fall regions	—	Visual understanding

Concept	Statement	Symbols Meaning	Key Notes
Constant Function	[f'(x) = 0]	—	Neither increasing nor decreasing
Polynomial	Continuous & differentiable everywhere	—	Smooth behavior ⭐

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