Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 33

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IMPORTANT

Earn 100

Check the validity of Rolle’s theorem for the function $y = x^{3} + 4 x^{2} - 7 x - 10$ in the interval $[- 1, 2]$

Important Questions on Continuity and Differentiability

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 34

Let

f (x) = 4 x^{3} - 3 x^{2} - 2 x + 1

, use Rolle's theorem to prove that there exist

c, 0 < c < 1

such that

f (c) = 0

.

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 35

If the equation

a_{0} x^{n} + a_{1} x^{n - 1} + ..... + a_{n - 1} x = 0

has a positive root

α

prove that the equation

n a_{0} x^{n - 1} + (n - 1) a_{1} x^{n - 2} + ...... + a_{n - 1} = 0

also has a positive root smaller that

α

.

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 36

f (x)

and

g (x)

are differentiable functions for

0 \leq x \leq 2

and such that

f (0) = 5, g (0) = 0, f (2) = 8, g (2) = 1

. Show that there exists a number

c

satisfying

0 < c < 2

and

f^{'} (c) = 3 g^{'} (c)

.

HARD

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 37

If

f, ϕ, ψ

are continuous in

[a, b]

and derivable in

]a, b[

then show that there is a value of

c

lying between

a

&

b

such that,

|\begin{matrix} f (a) & f (b) & f^{'} (c) \\ ϕ (a) & ϕ (b) & ϕ^{'} (c) \\ ψ (a) & ψ (b) & ψ^{'} (c) \end{matrix}| = 0

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 38

The function

y = \frac{2 - x^{2}}{x^{4}}

takes on equal values at the end-points of the interval

[- 1, 1]

. Make sure that the derivatives of this function does not vanish at any point of the interval

[- 1, 1]

, and explain this derivation from Rolle's theorem.

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Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 39

With the aid of Lagrange's formula prove the inequalities

\frac{a - b}{a} \leq \ln \frac{a}{b} \leq \frac{a - b}{b}

, for the condition

0 < b \leq a

.

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 40

Let

f : [a, b] \to R

be continuous on

[a, b]

and differentiable on

(a, b)

. If

f (a) < f (b)

, then show that

f^{'} (c) > 0

for some

c \in (a, b)

.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 27 - Continuity and Differentiability>EXERCISE-4>Q 41

Show that the function

f (x) = x^{n} + p x + q

cannot have more than two real roots if

n

is even and more than three if

n

is odd.