Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 22

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For the functions defined parametrically by the equations

$f (t) = x = \{\begin{matrix} 2 t + t^{2} \sin \frac{1}{t} & ; t \neq 0 \\ 0 & ; t = 0 \end{matrix}$ and

$g (t) = y = \{\begin{matrix} \frac{1}{t} \sin t^{2} & ; t \neq 0 \\ 0 & ; t = 0 \end{matrix}$ .

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Important Questions on Application of Derivatives

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 23

If

L_{T}, L_{N}, L_{S T}

and

L_{S N}

denote the lengths of a tangent, normal, sub-tangent and sub-normal, respectively, of a curve

y = f (x)

at a point

P (2009, 2010)

on it, then:

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 24

At the point

P (a, a^{n})

on the graph of

y = x^{n} (n \in N)

in the first quadrant a normal is drawn. The normal intersects the

Y

-axis at the point

(0, b)

. If

\lim_{a \to 0} b = \frac{1}{2},

then

n

equal

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 25

h (x) = 3 f (\frac{x^{2}}{3}) + f (3 - x^{2}), \forall x \in (- 3, 4)

, where

f^{''} (x) > 0, \forall x \in (- 3, 4)

, then

h (x)

is

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 26

If $f (x) = \{\begin{cases} - e^{- x} + k & , x \leq 0 \\ e^{x} + 1 & , 0 < x < 1 \\ e x^{2} + λ & , x \geq 1 \end{cases}$ is one-one and monotonically increasing $\forall x \in R,$ then

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 27

Let

f : R \to R, f (x) = x + \log_{e} (1 + x^{2})

. Then,

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 28

Equation

\frac{1}{{(x + 1)}^{3}} - 3 x + \sin x = 0

has

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 29

If

f (x) = a^{\{a^{|x|} s g n x\}}; g (x) = a^{[a^{|x|} \cdot sgn x]}

for

a > 0, a \neq 1

and

x \in R,

where

{.}

and

[.]

denotes the fractional part and integral part functions respectively, then which of the following statements hold good for the function

h (x)

, where

(\ln a) h (x) = (l n f (x) + l n g (x))

.

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Mathematics Crash Course JEE Advanced>Chapter 21 - Application of Derivatives>Exercise 1>Q 30

Let

ϕ (x) = {(f (x))}^{3} - 3 {(f (x))}^{2} + 4 f (x) + 5 x + 3 \sin x + 4 \cos x \forall x \in R

, where

f (x)

is a differentiable function

\forall x \in R

, then