Mathematics Times – July 2019

1. For


2. Find the smallest value of

x 0 , the smallest value of

 

4 8 13^2

6 1

x x

x

 

 is:

(a) 1 (b) 2 (c)

25

12

(d)

13

6

2

1 8

1

x

x

x x

 



if

x 1.

3. Find the smallest value of

y x x x x      (^1)  (^2)  (^3)  4 35.

4. Find the maximum and minimum values of

2

2

1

y x 1

x

  .

5. Find the greatest value of x y^2 ^2 if

x xy y x y^2     ^23  3 . x and yand

real.

6. Let x and y be real numbers satisfying the

equation x x y^2   2 4 5. Compute the

maximum value of x y 2.

7. The minimum value is mand the maximum

value is n of x xy y^2  ^2 for real x and y

are if x xy y^2   ^23. Find m n.

8. Find the greatest value^22
   9. Given that^22

x y x  2 if x and

y are real and satisfying:2 6x x y^2   ^20

x y x y   14 6 6, what is the

largest possible value that 3 4x y can

have?

(a) 72 (b) 73 (c) 74 (d) 75

10. Find the range of

1.Sol: Method 1

We seek the smallest value of

 

y x x   2 3 4 13 for

real x.

1. 
   

2. 
   

3. 
   

4. 
   

5. 
   

6. 
   

7. 
   

8. 
   

9.

10.

1.Sol: Method 1

 

 

(^242) 2 1 9
4 8 13
6 1 6 1
x x x x
y
x x
    
 
 
 
 
2
4 1 9
6 1
x
x
 


under the condition x 0.
Letting x z  1 , the expression whose
minimum we must determine may be written
as
4 9^2
6
z
y
z

.
Since x 0 and z 1 , we can multiply both
sides by 6z and obtain
4 9 6z zy^2   or 4 6 9 0z zy^2   .
Since z is real,

THE GREATEST AND SMALLEST VALUES