[SPLIT] Prove that for any n E N, a,b larger than or equal to 0,

[mathematicals]

I am stuck on this question:
Prove that for any n E N, a,b larger than or equal to 0,


(ab<sup>n )1/n+1 is less than or equal to a+nb/ n+1


Using the Am-Gm inequality, also if you could explain how you use it in this context it would be very hepful as well</sup>

 

[daon2]

I am stuck on this question:
Prove that for any n E N, a,b larger than or equal to 0,


(ab<sup>n )1/n+1 is less than or equal to a+nb/ n+1


Using the Am-Gm inequality, also if you could explain how you use it in this context it would be very hepful as well</sup>

Please create your own thread for your question, and in doing so, format your question correctly using superscripts or the caret symbol ("^") and use parentheses around exponents/numerators/denominators if necessary.

 

[mathematicals]

(ab^n)^[1 / (n+1)] <= (a + nb) / (n + 1)

Is that what you mean?

Proper bracketing (as Daon indicates) is IMPORTANT.

Yes, Sorry I was not sure how to do that on my computer...

 

[daon2]

Using the Arithmetic-Geometric mean inequality makes this problem trivial...

\(\displaystyle \sqrt[n+1]{x_1\cdot x_2\cdots x_{n+1}} \le \dfrac{x_1+x_2+\cdots + x_{n+1}}{n+1}\)

Now let \(\displaystyle x_1=a, x_2=x_3=\cdots=x_{n+1}=b\).