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Rolle'S Theorem, Definition, Geometrical Explanation, Important Topics For JEE 2024

Rolle's theorem : Rolle's theorem, in analysis, special case of the mean-value theorem of differential calculus. Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.
authorImageShrivastav 23 Jan, 2024
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Rolle's Theorem

Rolle'S Theorem : According to Rolle’s theorem if a function is continuous in the interval [ a , b ] and differentiable in the interval ( a , b ) and f ( a ) = f ( b ), then there exists at least one point c for which f ' ( c ) = 0 where c ε ( a , b ) Or There exists at least one point c at which slope of tangent is equal to zero.

Here as we can see function is continuous at [ a , b ] and differentiable at ( a , b ) and f ( a ) = f ( b ) = 0, so conditions of Rolle’s theorem are satisfied hence there must be a point for which f’ ( c ) = 0 as below.

In above graph there is a point c for which f’ ( c ) = 0 or slope of tangent is equal to zero.

Rolle'S Theorem Example 1

Rolle'S Theorem Example 1 : Verify Rolle’s theorem for the function f ( x ) = x 2 – 2 x +s 1 on [1, 3]

Sol: We have f ( x ) = x 2 – 2 x + 1 which is a polynomial function and polynomial functions are continuous and differentiable for x ε R , now to hold Rolle’s there must be c ε (1, 3) for which f ' ( c ) = 0 holds Let’s check.

f ' ( x ) = 2 x – 2, f ' ( x ) = 0 2 x – 2 = 0 ⇒ x = 1 ε (1, 3) Hence Rolle’s Theorem verified.

Rolle'S Theorem Example 2

Rolle'S Theorem Example 2 : Verify Rolle’s theorem for the function f ( x ) = | x – 6| on [–7, 7]

Sol:

As we can observe | x – 6| function is not differentiable at 6 hence Rolle’s theorem is not applicable for the given function.

Rolle'S Theorem Rapid Questions

  1. Verify Rolle’s theorem for the function x 2 -4 on [– 1, 1]?
  2. Verify Rolle’s theorem for the function x-5 +6 on [–3, 6]?

Explanation of Conditions in Rolle’s theorem

Why function must be continuous at [a, b] and differentiable at (a, b)? If a function is not continuous and differentiable then we can’t generalize the condition f ’( c ) = 0 as c is a general point between [a, b]

In above we can see there is no point c for which f ’( c ) = 0 exist.

Here f ’( c ) = 0 exist, Above two examples proof’s that Rolle’s theorem conditions are sufficient but not necessary.

Rolle'S Theorem Rapid Questions Example 1

Example 1 : Verify Rolle's theorem for the function in the interval [0, 5]

Sol:

As we can see in above graph function is not continuous at 1/3 hence Rolle’s theorem not applicable.

Rolle'S Theorem Rapid Questions Example 2

Example 2 : Verify Rolle’s Theorem for the function | x 2 – 3 x + 2 | in the interval [0, 6].

Sol:

As we can see in above graph function is not differentiable {1, 2} hence Rolle’s theorem not applicable.

Rapid Questions -

1. Verify Rolle’s Theorem for the function | x 2 – 5 x + 6 | in the interval [0, 7].

2. Verify Rolle’s Theorem for the function 15 x – 3in the interval [0, 3].

Why interval must be [ a , b ] for continuity? We may have three possibilities [ a , b ), ( a , b ], ( a , b )

Rolle'S Theorem Case-1 (a , b)

Here function is considered as continuous at [ a , b ) Let’s make a graph.

In above as we can see function is continuous at [ a , b ] and f ( a ) = f ( b ) also f ’( c ) = 0 Exist, Now let’s see second situation. In above we can see f ( a ) = f ( b ) also function is continuous and differentiable at [ a , b ) and ( a , b ) respectively. Still there is no point available [ a , b ) such that f ’( c ) = 0.

These two situations are possible, because Rolle’s theorem conditions are sufficient but not necessary.

Rolle'S Theorem Case-2 (a , b)

Case-2 [ a , b ) : In similar way [ a , b ) can also be explained.

Rolle'S Theorem Case-3 (a , b)

Case-3 ( a , b ) : In above as we can see function is continuous and differentiable at ( a , b ), f ( a ) = f ( b ) and also f ’( c ) = 0 exist Now let’s see next situation

In above as we can see function is continuous and differentiable at ( a , b ) and also f ( a ) = f ( b ) but there is no point c exist such that f ’( c ) = 0

These two situations are possible, because Rolle’s theorem conditions are sufficient but not necessary.

Why f ( a ) = f ( b ) ? As we know f’(c) represents slope of tangent and if starting and ending point has same values as well continuous and differentiable at [a, b] and ( a , b ) respectively, Then function must have to take at least one U turn except the case when line is parallel to x axis (then infinite points would exist).

In above we can see at [ a , b ] function is continuous and differentiable at ( a , b ) as well as f ( a ) = f ( b ) hence there are 3 points for which slope of tangent is zero.

In above function is continuous and differentiable at [a, b] and ( a , b ) respectively but f ( a ) ≠ f ( b ), still slope of tangent is zero, this is because Rolle’s theorem gives sufficient conditions not necessary.

When function would not take U turn?

If graph is a line parallel to x axis and continuous, differentiable at [a, b] and (a, b) respectively.

Rolle'S Theorem Illustrations

Q. 1. Verify Rolle’s Theorem for f ( x ) = sin2 x on [0, 2]

Sol : Since Sin2x is continuous and differentiable function in and f (0) .

So f ’( x ) = 0 ⇒ 2cos 2 x = 0, cos2 x = 0 ⇒ x = 4 Hence verified.

Q. 2. Verify Rolle’s Theorem for f(x)= 6 x – 4 x on [0, 6]

Sol: Since 6 x – 4 x is continuous and differentiable function in [0, 6] and f (0) = f (6) = 0.

So, f’( x ) = 0 ⇒ 6 – 8sin x = 0, sin x = 34 ⇒ x =.24

Hence verified.

Rapid Questions -

1. Verify Rolle’s Theorem for f ( x ) = – sin on [–1, 0]

2. Verify Rolle’s Theorem for f ( x ) = x 2 – 5 x + 4 on [1, 4]

Rolle's Theorem FAQs

Q.1: Does Rolle’s Theorem always hold for discontinuous functions on [a, b]?

Ans. No, as necessary condition to hold Rolle’s Theorem   is, function must be continuous. 

Q.2: Does Rolle’s Theorem always hold for differentiable functions on (a, b)?

Ans. Yes, Necessary condition to hold Rolle’s Theorem is, function must be differentiable on (a, b)

Q.3 : Does f'(c) = 0 represents slope of horizontal tangent?

Ans. Yes it represents slope of tangent which is horizontal.

Q. 4 :    There exists exactly one point between (a, b) for a function if Rolle’s Theorem hold on it?

Ans. No, there can be more than one U turn.

Q.5: Verify Rolle’s Theorem for the function f (x) = x(x + 1)(x + 2) on [0, 4].

Ans. No, obtained value is not in the interval.
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