SLecture5 Rolle's Theorem and Mean Value Theorem

MA 105 Calculus: Lecture 5Rolle’s Theorem and Mean Value Theorem

S. Sivaji Ganesh

Mathematics DepartmentIIT Bombay

August 3, 2009

S. Sivaji Ganesh (IIT Bombay) MA 105 Calculus: Lecture 5 August 3, 2009 1 / 20

Rolle’s theorem


Rolle’s theorem

We want to draw a graph of a function f with domain [0, 1].


Rolle’s theorem

We want to draw a graph of a function f with domain [0, 1]. Do thefollowing:


Rolle’s theorem


1 Mark the domain on x-axis


Rolle’s theorem


1 Mark the domain on x-axis2 Plot the values of f ,


Rolle’s theorem


1 Mark the domain on x-axis2 Plot the values of f , given by f (0) = f (1).


Rolle’s theorem


1 Mark the domain on x-axis2 Plot the values of f , given by f (0) = f (1). Choose this value to be

anything that you like.


Rolle’s theorem



anything that you like.3 Draw any ‘nice’ continuous curve joining these two points


Rolle’s theorem



anything that you like.3 Draw any ‘nice’ continuous curve joining these two points i.e.,

(0, f (0)) and (1, f (1))


Rolle’s theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function


Rolle’s theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Draw some more graphs


Rolle’s theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Draw some more graphs6 What are your observations?


Rolle’s theorem




(0, f (0)) and (1, f (1))


There is a point at which


Rolle’s theorem




(0, f (0)) and (1, f (1))


There is a point at which tangent is parallel to x-axis.


Rolle’s theorem




(0, f (0)) and (1, f (1))


There is a point at which tangent is parallel to x-axis. There may bemore than one.


Rolle’s theorem


Rolle’s theorem

Theorem (Rolle’s theorem)

Let f be a function that satisfies the following three hypotheses:


Rolle’s theorem


Let f be a function that satisfies the following three hypotheses:1 f is continuous on the closed interval [a, b].


Rolle’s theorem


Let f be a function that satisfies the following three hypotheses:1 f is continuous on the closed interval [a, b].2 f is differentiable on the open interval (a, b).


Rolle’s theorem


Let f be a function that satisfies the following three hypotheses:1 f is continuous on the closed interval [a, b].2 f is differentiable on the open interval (a, b).3 f (a) = f (b).


Rolle’s theorem



Then


Rolle’s theorem



Then there is a number c in the open interval (a, b)


Rolle’s theorem



Then there is a number c in the open interval (a, b) such that


Rolle’s theorem



Then there is a number c in the open interval (a, b) such that f ′(c) = 0.


Rolle’s theorem




Note: All our examples illustrate Rolle’s theorem.


Rolle’s theorem




Note: All our examples illustrate Rolle’s theorem.Proof: is an application of EVT and Fermat’s theorem.


Rolle’s theorem





1 If f is a constant function


Rolle’s theorem





1 If f is a constant function i.e., f (x) = f (a) for every x ∈ [a, b],


Rolle’s theorem





1 If f is a constant function i.e., f (x) = f (a) for every x ∈ [a, b], sucha c exists.


Rolle’s theorem





1 If f is a constant function i.e., f (x) = f (a) for every x ∈ [a, b], sucha c exists. why?


Rolle’s theorem






2 If f is not a constant,


Rolle’s theorem






2 If f is not a constant, then at least one of the following holds.


Rolle’s theorem






2 If f is not a constant, then at least one of the following holds.(i) The graph of f goes above the line y = f (a)


Rolle’s theorem






2 If f is not a constant, then at least one of the following holds.(i) The graph of f goes above the line y = f (a) i.e., f (x) > f (a) for

some x ∈ (a, b).


Rolle’s theorem







some x ∈ (a, b).(ii) The graph of f goes below the line y = f (a)


Rolle’s theorem







some x ∈ (a, b).(ii) The graph of f goes below the line y = f (a) i.e., f (x) < f (a) for

some x ∈ (a, b).S. Sivaji Ganesh (IIT Bombay) MA 105 Calculus: Lecture 5 August 3, 2009 7 / 20

Rolle’s theorem


Rolle’s theorem

Proof of Rolle’s theorem (contd.):


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint)


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b.


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So?


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b).


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it c.


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it c.That is, global maximum on [a, b] is actually a local maximum


Rolle’s theorem


2(i) If the graph of f goes above the line y = f (a), then where is theglobal maximum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it c.That is, global maximum on [a, b] is actually a local maximum ByFermat’s theorem f ′(c) = 0.


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint)


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b.


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So?


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b).


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it d .


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it d .That is, global minimum on [a, b] is actually a local minimum


Rolle’s theorem



2(ii) If the graph of f goes below the line y = f (a), then where is theglobal minimum attained? (EVT guarantees existence of such apoint) It cannot be at a or b. So? it must inside (a, b). call it d .That is, global minimum on [a, b] is actually a local minimum ByFermat’s theorem f ′(d) = 0.


Rolle’s theorem




This finishes the proof of Rolle’s theorem.


Rolle’s theorem




This finishes the proof of Rolle’s theorem.Graph of any function satisfying the hypothesis of Rolle’s theorem beremembered as


Rolle’s theorem




This finishes the proof of Rolle’s theorem.Graph of any function satisfying the hypothesis of Rolle’s theorem beremembered as a graph having geometry of Rolle’s theorem


Mean Value Theorem


Mean Value Theorem

We want to draw a graph of a function f with domain [0, 1].


Mean Value Theorem



Mean Value Theorem


1 Mark the domain on x-axis


Mean Value Theorem


1 Mark the domain on x-axis2 Mark the points (0, f (0)) and (1, f (1)).


Mean Value Theorem


1 Mark the domain on x-axis2 Mark the points (0, f (0)) and (1, f (1)). Let f (0), f (1) be of your

choice.


Mean Value Theorem



choice.3 Draw any ‘nice’ continuous curve joining these two points


Mean Value Theorem



choice.3 Draw any ‘nice’ continuous curve joining these two points i.e.,

(0, f (0)) and (1, f (1))


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Join the two points (0, f (0)) and (1, f (1)) by a straight line


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Join the two points (0, f (0)) and (1, f (1)) by a straight line6 What are your observations?


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Join the two points (0, f (0)) and (1, f (1)) by a straight line6 What are your observations?7 Is there a point on the graph


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Join the two points (0, f (0)) and (1, f (1)) by a straight line6 What are your observations?7 Is there a point on the graph where the tangent is


Mean Value Theorem




(0, f (0)) and (1, f (1))

4 We have finished defining our function5 Join the two points (0, f (0)) and (1, f (1)) by a straight line6 What are your observations?7 Is there a point on the graph where the tangent is parallel to the

straight line which was drawn?


Mean Value Theorem




(0, f (0)) and (1, f (1))



There is such a point.


Mean Value Theorem




(0, f (0)) and (1, f (1))



There is such a point.There may be more than one.


Mean value theorem


Mean Value Theorem


Mean Value Theorem

This theorem is due to J.-L. Lagrange.


Mean Value Theorem


Theorem (Mean value theorem)

Let f be a function that satisfies the following hypotheses:


Mean Value Theorem



Let f be a function that satisfies the following hypotheses:1 f is continuous on the closed interval [a, b].


Mean Value Theorem



Let f be a function that satisfies the following hypotheses:1 f is continuous on the closed interval [a, b].2 f is differentiable on the open interval (a, b).


Mean Value Theorem




Then


Mean Value Theorem




Then there is a number c in the open interval (a, b)


Mean Value Theorem






Mean Value Theorem





f ′(c) =f (b) − f (a)

b − a.


Mean Value Theorem





f ′(c) =f (b) − f (a)

b − a.

or, equivalently,f (b)− f (a) = f ′(c)(b − a).


Mean Value Theorem





f ′(c) =f (b) − f (a)

b − a.

or, equivalently,f (b)− f (a) = f ′(c)(b − a).

Note: Our examples illustrate Mean value theorem.


Mean Value Theorem


Mean Value Theorem

Questions:


Mean Value Theorem

Questions:1 Can we prove Rolle’s theorem using the Mean value theorem?


Mean Value Theorem


yes!


Mean Value Theorem


yes! Easy proof.


Mean Value Theorem


yes! Easy proof.2 We deduce Mean value theorem from Rolle’s theorem here.


Mean Value Theorem


yes! Easy proof.2 We deduce Mean value theorem from Rolle’s theorem here.3 Therefore, the statements of the two theorems are “one and the

same” or “equivalent”.


Mean Value Theorem


Mean Value Theorem

Proof of Mean value theorem:


Mean Value Theorem

Proof of Mean value theorem: The strategy is to define a newfunction φ(x)


Mean Value Theorem

Proof of Mean value theorem: The strategy is to define a newfunction φ(x) satisfying the hypothesis of Rolle’s theorem.


Mean Value Theorem

Proof of Mean value theorem: The strategy is to define a newfunction φ(x) satisfying the hypothesis of Rolle’s theorem. Theconclusion of Rolle’s theorem for φ should yield the conclusion of MVTfor f .


Mean Value Theorem


1 Define φ on [a, b] by

φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ?


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ on [a, b]?


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ on [a, b]? Yes!


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ on [a, b]? Yes!3 Rolle’s theorem asserts the existence of c ∈ (a, b)


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ on [a, b]? Yes!3 Rolle’s theorem asserts the existence of c ∈ (a, b) such that

φ′(c) = 0.


Mean Value Theorem



φ(x) = f (x) − f (a) − f (b) − f (a)

b − a(x − a).

What is the graph of φ? Does it have Rolle’s theorem geometry?2 Can we apply Rolle’s theorem to φ on [a, b]? Yes!3 Rolle’s theorem asserts the existence of c ∈ (a, b) such that

φ′(c) = 0.4 This finishes the proof of MVT.


On the proof of Mean value theorem



What is φ in the following graph?


Mean value theorem: Applications



1 If f ′(x) = 0 for all x ∈ (a, b), then



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b)



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b) and x1 < x2.



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b) and x1 < x2. ApplyMVT for f on [x1, x2].



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b) and x1 < x2. ApplyMVT for f on [x1, x2]. We get f (x1) = f (x2).



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b) and x1 < x2. ApplyMVT for f on [x1, x2]. We get f (x1) = f (x2). Since this is true forany x1 and x2 in (a, b) with x1 < x2,



1 If f ′(x) = 0 for all x ∈ (a, b), then f is constant on (a, b).Proof Let x1 and x2 be any two points in (a, b) and x1 < x2. ApplyMVT for f on [x1, x2]. We get f (x1) = f (x2). Since this is true forany x1 and x2 in (a, b) with x1 < x2, we conclude f is a constantfunction.




2 If f ′(x) = g′(x) for all x ∈ (a, b), then




2 If f ′(x) = g′(x) for all x ∈ (a, b), then f − g is constant on (a, b).




2 If f ′(x) = g′(x) for all x ∈ (a, b), then f − g is constant on (a, b).That is, f (x) = g(x) + C for some real number C.





Let f be a function defined by f (x) = x|x| for x 6= 0.





Let f be a function defined by f (x) = x|x| for x 6= 0. What is f ′?






Constant function 0.






Constant function 0. Is f constant?






Constant function 0. Is f constant? No.






Constant function 0. Is f constant? No. Does it contradict (1) above?


Problems


Problems

1 Does there exist a function f such that f (0) = −1, f (2) = 4, andf ′(x) ≤ 2 for all x?


Problems


2 Show that the equation 2x − 1 − sin x = 0 has exactly one root.


Problems


2 Show that the equation 2x − 1 − sin x = 0 has exactly one root.3 Show that the equation x3 − 15x + c = 0 has at most one root in

[−2, 2].


Problems



[−2, 2].4 Show that the equation x4 + 4x + c = 0 has at most two real roots.


Problems



[−2, 2].4 Show that the equation x4 + 4x + c = 0 has at most two real roots.5 If f (1) = 10, and f ′(x) ≥ 2 for 1 ≤ x ≤ 4, how small can f (4)

possibly be?


Problems



[−2, 2].4 Show that the equation x4 + 4x + c = 0 has at most two real roots.5 If f (1) = 10, and f ′(x) ≥ 2 for 1 ≤ x ≤ 4, how small can f (4)

possibly be?6 Show that

√1 + x < 1 + 1

2x if x > 0.


SLecture5 Rolle's Theorem and Mean Value Theorem

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