A point P is given on the circumference of a circle of radius r . Chord Q R is parallel to the tangent at P . Determine the maximum possible area of the ∆ P Q R .

Given: A circle with radius r and a chord Q R , which is parallel to the tangent at P . So, Area of ∆ P Q R , A = 1 2 × Base × Height Here, Base = Q R and Height = P N ⇒ A = 1 2 × Q R × P N In ∆ P Q R , P N is perpendicular to Q R . So, P N = P C + C N , as P C = r (radius) In ∆ C N R , C N = r cos θ Therefore, P N = r + r cos θ Also, Q R = Q N + N R where Q N = r sin θ and N R = r sin θ ⇒ Q R = 2 r sin θ So, Area of ∆ P Q R , A = 1 2 Q R × P N = 1 2 × 2 r sin θ ( r + r cos θ ) = 1 2 r 2 [ 2 sin θ + 2 sin θ cos θ ] Therefore A ( θ ) = 1 2 r 2 [ 2 sin θ + sin 2 θ ] Differentiating, A ( θ ) = 1 2 r 2 [ 2 sin θ + sin 2 θ ] with respect to θ , we get ⇒ A ' ( θ ) = 1 2 r 2 [ 2 cos θ + 2 cos 2 θ ] ⇒ A ' ( θ ) = r 2 [ cos θ + cos 2 θ ] Put A ' ( θ ) = 0 in order to get critical points for maxima and minima. ⇒ r 2 [ cos θ + cos 2 θ ] = 0 ⇒ cos θ + 2 cos 2 θ - 1 = 0 ⇒ 2 cos 2 θ + 2 cos θ - cos θ - 1 = 0 2 cos θ ( cos θ + 1 ) - 1 ( cos θ + 1 ) = 0 ⇒ ( 2 cos θ - 1 ) ( cos θ + 1 ) = 0 Therefore, 2 cos θ - 1 = 0 o r cos θ + 1 = 0 ⇒ cos θ = 1 2 or cos θ = - 1 ⇒ θ = π 3 or θ = π But in any triangle one angle cannot be π . So, θ = π 3 is the only critical point. Differentiating A ' ( θ ) = r 2 [ cos θ + cos 2 θ ] with respect to θ ⇒ A '' ( θ ) = r 2 [ - sin θ - 2 sin 2 θ ] ⇒ A ' ' π 3 = r 2 - sin π 3 - 2 sin 2 · π 3 ⇒ A '' π 3 = r 2 - 3 2 - 2 · 3 2 < 0 As we know that from the theory of maxima and minima, if f ' ' x is negative on a critical point, then at that point, value of the function is maximum. Therefore, θ = π 3 is a point of maxima. So, area of ∆ P Q R , A = 1 2 × 2 r sin θ ( r + r cos θ ) when θ = π 3 . Area of ∆ P Q R , A = 1 2 × 2 r sin π 3 r + r cos π 3 = 2 r 2 3 2 r 1 + 1 2 = r 2 3 2 3 2 = 3 3 r 2 4 Hence, the maximum possible area of the ∆ P Q R is 3 3 4 r 2 .

HARD

JEE Main

IMPORTANT

Earn 100

Find the range of parameter $b$ , for which the function $f (x)$ is entirely increasing or decreasing for all values of $x$ where,
$f (x) = \int_{0}^{x} (b t^{2} + b + c o s t) d t$

Important Questions on Monotonicity, Maxima and Minima

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 17

A box is made with a square base and an open top. The area of the material used is

192 s q . c m

. Find the dimensions of the box, if its volume is maximum.

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 18

A closed rectangular box with a square base is to be made to hold

1000

cubic feet. The cost of the material per sq foot for the bottom is

15

paise, for the top

25 p a i s e

and for the sides

20 p a i s e

. The labour charge for making the box is

₹ 3 / -

.Find the dimensions of the box when the cost is minimum.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 19

A running track of

440

yards is to be laid out encircling a football field the shape of which is rectangle with a semi-circle at each end. If the area of the rectangular position is to be maximum find the lengths of its sides.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 20

Show that a right circular conical tent of volume

V

will require the least amount of canvas to make it if it's height is

\sqrt{2}

times the radius of its base.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 21

Two corridors of width

a

and

b

are at right angles. Show that the length of the longest pipe that can be passed round the corner horizontally is

{(a^{2 / 3} + b^{2 / 3})}^{3 / 2} .

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 22

A point

P

is given on the circumference of a circle of radius

(i v)

. Chord

Q R

is parallel to the tangent at

P

. Determine the maximum possible area of the

∆ P Q R

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 23

Two cars are travelling along two roads which cross each other at right angles at

A

. One car is travelling towards

A

, at a speed of

21

miles per hour while the other is travelling towards it at a speed of

28

miles per hour. If initially, their distance from

A

are,

1500

feet and

2100

feet respectively, prove that the least distance between them is

60

feet.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 8 - Monotonicity, Maxima and Minima>Proficiency in 'Monotonicity, Maxima and Minima' Exercise 2>Q 24

Two roads

O A

O B

intersect at an angle of

60^{o}

. A car driver approaches

O

from

A

, where

A O = 800 m

at a uniform speed of

20 m / s

. Simultaneously, a runner starts running from

O

towards

B

at a uniform speed of

5 m / s

. Find the time when the car and the runner are closest.

Find the range of parameter b, for which the function fx is entirely increasing or decreasing for all values of x where, fx=∫0xbt2+b+costdt

Important Questions on Monotonicity, Maxima and Minima

Learn and Prepare for any exam you want !

Find the range of parameter $b$ , for which the function $f (x)$ is entirely increasing or decreasing for all values of $x$ where,
$f (x) = \int_{0}^{x} (b t^{2} + b + c o s t) d t$