# Geometrical Explanation of Rolle's Theorem

## Trending Questions

**Q.**Show that f : R $\to $ R, given by f(x) = x $-$ [x], is neither one-one nor onto.

**Q.**

What is constant function? In relation and function chapter

**Q.**The function f(x)=x(x+3)e−(1/2)x satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

- 0
- -1
- -2
- -3

**Q.**The absolute maximum value of a function f given by

f(x)=2x3−15x2+36x+1 on the interval [1, 5] is

**Q.**

On which of the following functions can we apply LMVT in the interval [-2, 2] ?

Y=|x|

y = e

^{x}y=x

^{3}y=[x]

y=cosx

**Q.**

Does there exist a function which is continuous every where but not differentiable at exactly two points? Justify your answer.

**Q.**

Find
all the points of discontinuity of *f
*defined by.

**Q.**An example of function which is continuous everywwhere but not differentiable at exactly at two points is

- f(x)=|x|+|x−1|
- f(x)=|x+1|+|x|−|x−3|
- f(x)=|x+1|−|x−3|+|x−7|
- None

**Q.**

The above figure is the graph of a continuous and differentiable function y = f(x). Between point A & B the function has its derivative zero at how many points -

3

4

5

None of these

**Q.**Rolle's theorem is not applicable to the function f(x) = |x| defined on [–1, 1] because

[AISSE 1986; MP PET 1994, 95]

f is not differentiable on (–1, 1)

f(−1)≠f(1)

- f(−1)=f(1)≠0
f is not continuous on [ –1, 1]

**Q.**Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1].

**Q.**The function f(x)=x(x+3)e−(1/2)x satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

- \N
- -1
- -2
- -3

**Q.**The function f : R → R defined by $f\left(x\right)={2}^{x}+{2}^{\left|x\right|}\mathrm{is}$

(a) one-one and onto

(b) many-one and onto

(c) one-one and into

(d) many-one and into

**Q.**Statement 1: Rolle's theorem is not applicable to f(x)=(x−1)|x−1| on [1, 2].

Statement 2:|x−1| is not differentiable at x=1.

- Only statement 1 is true
- Only statement 2 is true
- Both are true
- Both are false

**Q.**The figure given is the graph of a continuous and differentiable function y = f(x). Between point A & B the function has its derivative zero at how many points -

- 4
- 5
- 3
- None of these

**Q.**

Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), *a* = 4

**Q.**Given that the derivative f′(a) exists. Indicate which of the following statements (s) is/are always true-

- f′(a)=limh→af(h)−f(a)h−a
- f′(a)=limh→0f(a)−f(a−h)h
- f′(a)=limt→0f(a+2t)−f(a)t
- f′(a)=limt→0f(a+2t)−f(a+t)2t

**Q.**The figure given is the graph of a continuous and differentiable function y = f(x). Between point A & B the function has its derivative zero at how many points -

- 3
- 4
- 5
- None of these

**Q.**What does it mean when slope comes out as a no. divided by zero?

What can be said about the lines passing through these points (3, 4) and (3, 6) ?

6-4/3-3 = 2/0

**Q.**Examine if Rolles theorem is applicable to any one of the following functions.

(i) f (x) = [x] for x ∈ [5, 9]

(ii) f (x) = [x] for x ∈ [−2, 2]

Can you say something about the converse of Rolles Theorem from these functions?

**Q.**Rolle's theorem is not applicable to the function f(x) = |x| defined on [–1, 1] because

[AISSE 1986; MP PET 1994, 95]

f is not differentiable on (–1, 1)

f(−1)≠f(1)

- f(−1)=f(1)≠0
f is not continuous on [ –1, 1]

**Q.**

Find
all the points of discontinuity of *f
*defined by.

**Q.**

Find the
derivative of cos *x* from first principle.

**Q.**

Determine if *f* defined by

is a continuous function?

**Q.**If f(x) and g(x) both are discontinuous at any point, then show that their composition may be differentiable at that point.

**Q.**

Determine if *f*
defined by

is a continuous function?

**Q.**The function f(x)=x(x+3)e−(1/2)x satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

- -3
- -2
- 0
- -1

**Q.**The figure given is the graph of a continuous and differentiable function y = f(x). Between point A & B the function has its derivative zero at how many points -

- 3
- 4
- 5
- None of these

**Q.**

Discuss
the continuity of the function *f*,
where *f* is
defined by