Rolle’s Theorem & Lagrange’s Mean Value Theorem — Complete Formula

Rolle’s Theorem

Concept	Formula / Condition	Symbols Meaning	Key Notes / Tricks
Continuity	[f(x) \text{ is continuous on } [a, b]]	[f(x)] = function, [a, b] = interval	Must be continuous ⭐
Differentiability	[f(x) \text{ is differentiable on } (a, b)]	—	No sharp points ⚠️
Equal Values	[f(a) = f(b)]	—	Endpoints same ⭐

Concept	Formula	Symbols Meaning	Key Notes
Result	[\exists , c \in (a, b) \text{ such that } f'(c) = 0]	[c] = some point in interval	Horizontal tangent ⭐

💡 Memory Hint:
Same endpoints → flat point in between

Concept	Interpretation	Symbols Meaning	Key Notes
Meaning	Tangent parallel to x-axis exists	—	Slope = 0 ⭐

Concept	Formula / Condition	Symbols Meaning	Key Notes
Continuity	[f(x) \text{ is continuous on } [a, b]]	—	Required ⭐
Differentiability	[f(x) \text{ is differentiable on } (a, b)]	—	Must hold ⭐

Concept	Formula	Symbols Meaning	Key Notes
LMVT Result	[\exists , c \in (a, b) \text{ such that } f'(c) = \dfrac{f(b) – f(a)}{b – a}]	[c] = some point	Very important ⭐

💡 Memory Hint:
Instantaneous slope = average slope

Concept	Interpretation	Symbols Meaning	Key Notes
Meaning	Tangent parallel to chord exists	—	Key idea ⭐

Concept	Relation	Symbols Meaning	Key Notes
Special Case	Rolle’s is special case of LMVT when [f(a) = f(b)]	—	Then slope = 0 ⭐

💡 Memory Hint:
LMVT → general
Rolle → special case

Concept	Formula / Use	Symbols Meaning	Key Notes
Proving Roots	Show [f'(c) = 0] exists	—	Root-based questions ⭐
Inequalities	Use LMVT to bound values	—	Advanced ⭐
Function Behavior	Use derivative condition	—	Increasing/decreasing

Concept	Statement	Symbols Meaning	Key Notes
Converse Not True	Conclusion does not guarantee conditions	—	Important ⚠️
Condition Failure	If any condition fails → theorem not applicable	—	Check carefully ⭐

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