Chapter 18 Maxima and Minima Ex 18.1

Question 1

Solution 1

Question 2

Find the maximum and minimum values, if any, without using derivatives of the following function given by f(x) = -(x-1)² + 2 on R.Solution 2

Question 3

Solution 3

Question 4

Find the maximum and minimum values, if any, without using derivatives of the following function given by h(x) = sin(2x) + 5 on R.Solution 4

Question 5

Find the maximum and minimum values, if any,  usingwithout derivatives of the following function given by  on R.Solution 5

Question 6

Solution 6

Question 7

Find the maximum and minimum values, if any, without using derivatives of the following function given by  on R.Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 18 Maxima and Minima Ex. 18.2

Question 1

Solution 1

Question 2

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) =x³ – 3xSolution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) = sinx – cos x, 0 < x < 2πSolution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 18 Maxima and Minima Ex. 18.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3

Solution 3

Question 4

Show that the function given by f(x)= has maximum value at x = e.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

If f(x) = x³ + ax² + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.Solution 7

Question 8

Prove that  has maximum value at Solution 8

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

Differentiating f'(x) w.r.t x, we get

At 

Clearly, f”(x) < 0 at 

Thus,  is the maxima.

Hence, f(x) has maximum value at .

Chapter 18 Maxima and Minima Ex. 18.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Find both the absolute maximum and absolute minimum of 3x⁴ – 8x³ + 12x² – 48x + 25 on the interval [0, 3] Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 18 Maxima and Minima Ex. 18.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Divide 15 into two parts such that the square of one multiplied with the cube of the other is maximum.Solution 4

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

A square piece of tin of side 18 cm is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? Also, find this maximum volume.Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

An isosceles triangle of vertical angle 2θ is inscribed in a circle radius a. show that the area of the triangle is maximum when Solution 22

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its semi – circular ends is π : ( π + 2) Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 45

Solution 45

Question 5

Amongst all open (from the top) right circular cylindrical boxes of volume 125π cm³, find the dimensions of the box which has the least surface area.Solution 5

Let r and h be the radius and height of the cylinder.

Volume of cylinder  

 … (i)

Surface area of cylinder  

From (i), we get

Question 23

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is  Solution 23

Let  be an isosceles triangle with AB = AC.

Let  

Here, AO bisects 

Taking O as the centre of the circle, join OE, OF and OD such that

OE = OF = OD = r (radius)

Now, 

In 

Similarly, AF = r cot x

In 

As OB bisect  we have

In 

Similarly, BD = DC = CE = 

We have, perimeter of 

P = AB + BC + CA

 = AE + EC + BD + DC + AF + BF

Differentiating w.r.t x, we get

Taking 

As 

Therefore,  is an equilateral triangle.

Taking second derivative of P, we get

At 

Therefore, the perimeter is minimum when 

Least value of P