Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 10

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If the set of all values of the parameter $a$ for which the function $f (x) = \sin 2 x - 8 (a + 1) \sin x + (4 a^{2} + 8 a - 14) x$ increases for all $x \in R$ and has no critical points for all $x \in R$ , is $(- \infty, - m - \sqrt{n}) \cup (\sqrt{n}, \infty)$ then $(m^{2} + n^{2})$ is (where $m and n$ are prime numbers):

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Important Questions on Application of Derivatives

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 11

If

f (x) = \frac{x^{2}}{2 - 2 \cos x}; g (x) = \frac{x^{2}}{6 x - 6 \sin x}

where

0 < x < 1

, then

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 12

Let

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

. If the equation

f (x) = k

has exactly one positive and one negative solution, then the value of

k

equals to

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 13

Let

f (x)

be a polynomial of degree four having extreme values at

x = 1

and

x = 2

. If

\lim_{x \to 0} [1 + \frac{f (x)}{x^{2}}] = 3

, then

f (2)

is equal to

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 14

Rectangles are inscribed in a circle of radius

r

. The dimensions of the rectangle which has the maximum area, are

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 15

The minimum value of

\frac{\tan (x + \frac{π}{6})}{\tan x}

is

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 16

A point on the hypotenuse of a triangle is at distances

a

and

b

from the sides of the triangle, then the minimum length of the hypotenuse is

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 17

If the vertices of a triangle are

(0, 0), (x, \cos x)

and

(\sin^{3} x, 0)

, where

0 < x < \frac{π}{2}

. Then, the maximum area for such a triangle (in sq.units), is

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Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 18

The minimum value of

α

for which the equation

\frac{4}{\sin x} + \frac{1}{1 - \sin x} = α

has at least one solution in

(0, \frac{π}{2})

is______.