Positive Numbers a and b

[nycmathguy]

David Cohen

How is this done? Any ideas? Seeking hints for parts A and B.

 

[MathLover1]

Suppose that and are positive numbers whose sum is 1
(a) Find the maximum possible value of the product ab.
(b) Prove that

Given a and b are positive numbers and their sum is 1

So,
a+b=1
b=1-a.............……(1)

(a)

We have to find the maximum possible value of the product ab.

ab
=a(1-a)........from equation (1)
=a-a^2

Since the equation is that of a downward parabola therefore the input a that yields the maximum value of the equation will be

a=-b/2a
=-1/(2(-1))
=1/2

hence
b=1-1/2
b=1/2

Therefore the maximum value of

ab=(1/2)(1/2)=1/4


b) Prove that (1+1/a)(1+1/b)>=9

which is true

 

[nycmathguy]

Suppose that and are positive numbers whose sum is 1
(a) Find the maximum possible value of the product ab.
(b) Prove that

Given a and b are positive numbers and their sum is 1

So,
a+b=1
b=1-a.............……(1)

(a)

We have to find the maximum possible value of the product ab.

ab
=a(1-a)........from equation (1)
=a-a^2

Since the equation is that of a downward parabola therefore the input a that yields the maximum value of the equation will be

a=-b/2a
=-1/(2(-1))
=1/2

hence
b=1-1/2
b=1/2

Therefore the maximum value of

ab=(1/2)(1/2)=1/4


b) Prove that (1+1/a)(1+1/b)>=9

which is true

I never would have been able to figure this out.