show that for all a ≥ 1 , the function f defined by f x = 3 sin x - cos x - 2 a x + b is decreasing on R .

Given function is f x = 3 sin x - cos x - 2 a x + b . . . . . . . . . . 1 Putting 3 = r cos α ; - 1 = r sin α in 1 On squaring and adding ; we have 3 2 + - 1 2 = r cos α 2 + r sin α 2 ⇒ 3 + 1 = r 2 cos 2 α + r 2 sin 2 α ⇒ 4 = r 2 cos 2 α + sin 2 α ⇒ 4 = r 2 · 1 ∵ cos 2 A + sin 2 A = 1 ⇒ r = 2 and on dividing; we have r sin α r cos α = - 1 3 ⇒ tan α = tan - π 6 ⇒ α = - π 6 Now, put 3 = 2 cos - π 6 ; - 1 = 2 sin - π 6 in equation 1 ; we have f x = 2 cos - π 6 sin x + 2 sin - π 6 cos x - 2 a x + b = 2 sin x cos - π 6 + cos x sin - π 6 - 2 a x + b = 2 sin x - π 6 - 2 a x + b Differentiating both sides w.r.t. x , we have d d x f x = d d x 2 sin x - π 6 - 2 a x + b f ' x = 2 d d x sin x - π 6 - 2 a d d x x + d d x b = 2 cos x - π 6 - 2 a · 1 + 0 = 2 cos x - π 6 - 2 a Now, - 1 ≤ cos x - π 6 ≤ 1 ⇒ - 2 ≤ 2 cos x - π 6 ≤ 2 ⇒ - 2 - 2 a ≤ 2 cos x - π 6 - 2 a ≤ 2 - 2 a ≤ 0 [if a ≥ 1 ⇒ 2 a ≥ 2 ⇒ 2 - 2 a ≤ 0 ] ⇒ f ' x ≤ 0 Thus, f x be a decreasing function if a ≥ 1 .

MEDIUM

12th ICSE

IMPORTANT

Earn 100

Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3 \sqrt{3} cm$ .

Important Questions on Applications of Derivatives

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.8

A cone is inscribed in a sphere of radius

12 cm

. If the volume of the cone is maximum, find its height.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.8

A manufacturer plans to construct a cylindrical can to hold one cubic meter of oil. If the cost of constructing top and bottom of the can is twice the cost of constructing the side, what are the dimensions of the most economical can?

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

The radius of the base of a right circular cone is increasing at the rate of

3

cm per minute and the altitude is decreasing at the rate of

4

cm per minute. Find the rate of change of total surface area of the cone when its radius is

7

cm and altitude is

24

cm is of the form of

a π {cm}^{2} / \min

, then the value of

a

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

Find the equation of normal to the curve

y = x + \frac{2}{x}

at the point where abscissa is 2. If this normal meets the axes in points

A

and

B

. Length of

AB

\frac{a}{b} \sqrt{5} units

, where

r = 3 \sqrt{3} cm

are smallest positive integers. Find

a + b

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

show that for all

a \geq 1,

the function

f

defined by

f (x) = \sqrt{3} \sin x - \cos x - 2 a x + b

is decreasing on

R .

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

The sum of greatest and the least value of

f (x) = x^{2} \log x

[1, e]

e^{2} + k

, then find the value of

k

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

A lighthouse is located at

A, 2

km off-shore from the nearest point

O

on a straight beach and a shop is located at

B

on the beach at a distance of

4

km from

O .

Housekeeper can row at the speed of

4

km/h and can walk at the rate of

6

km/h and he plan to reach the shore so as to cover the distance to the shop in the least possible time. The man should reach the shore at the point whose distance from the point

O

\sqrt{a} km

. Find

a

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 1>Chapter 7 - Applications of Derivatives>EXERCISE 7.9

A cone is circumscribed about a sphere of radius

R .

Show that the volume of the cone is minimum if its height is

4 R .

Find the maximum volume of the cylinder which can be inscribed in a sphere of radius 33 cm.

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3 \sqrt{3} cm$ .