Geometrical Explanation of Rolle's Theorem

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 $f\left(x\right)=x(x+3)e^{-\left(1/2\right)x}$ satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

1. 0
2. -1
3. -2
4. -3

Q. The absolute maximum value of a function $f$ given by
$f\left(x\right)=2x^3-15x^2+36x+1$ on the interval $[1,5]$ is

Q.

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

1. Y=|x|

2. y = e^x

3. y=x³

4. y=[x]

5. 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

1. $f\left(x\right)=|x|+|x-1|$
2. $f\left(x\right)=|x+1|+|x|-|x-3|$
3. $f\left(x\right)=|x+1|-|x-3|+|x-7|$
4. 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 -

1. 3

2. 4

3. 5

4. 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]

1. f is not differentiable on (–1, 1)

2. $f(-1)\ne f\left(1\right)$

3. $f(-1)=f\left(1\right)\ne 0$
4. 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 $f\left(x\right)=x(x+3)e^{-\left(1/2\right)x}$ satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

1. \N
2. -1
3. -2
4. -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 $f\left(x\right)=(x-1)|x-1|$ on $[1,2].$
Statement $2:|x-1|$ is not differentiable at $x=1$.

1. Only statement $1$ is true
2. Only statement $2$ is true
3. Both are true
4. 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 -

1. 4
2. 5
3. 3
4. 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 $f^{\prime }\left(a\right)$ exists. Indicate which of the following statements (s) is/are always true-

1. $f^{\prime }\left(a\right)={lim}_{h\to a}\frac{f\left(h\right)-f\left(a\right)}{h-a}$
2. $f^{\prime }\left(a\right)={lim}_{h\to 0}\frac{f\left(a\right)-f(a-h)}{h}$
3. $f^{\prime }\left(a\right)={lim}_{t\to 0}\frac{f(a+2t)-f\left(a\right)}{t}$
4. $f^{\prime }\left(a\right)={lim}_{t\to 0}\frac{f(a+2t)-f(a+t)}{2t}$

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 -

1. 3
2. 4
3. 5
4. 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]

1. f is not differentiable on (–1, 1)

2. $f(-1)\ne f\left(1\right)$

3. $f(-1)=f\left(1\right)\ne 0$
4. 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 $f\left(x\right)$ and $g\left(x\right)$ 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 $f\left(x\right)=x(x+3)e^{-\left(1/2\right)x}$ satisfies all the conditions of Rolle's theorem in [–3, 0]. The value of c is

1. -3
2. -2
3. 0
4. -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 -

1. 3
2. 4
3. 5
4. None of these

Q.

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