Solving Inequalities with c and n: How-To Guide

[discy]

Homework Statement

4. Give a c > 0 and an integer n0 ≥ 1 such that, for all n ≥ n0.

b. 16n log (n²) ≤ cn²

The answer (from the sheet) is c = 32

Homework Equations

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The Attempt at a Solution

When I attempt to solve such an equation I start at n=1, then go to n=2. but that way I get the answer c=17.

I understand this is kind of a brute force attack. I would like to know what the proper way would be to solve this equation.

 

[obafgkmrns]

You can fiddle with the equation a little to get

32 log(n)/n ≤ c

Then you can graph it to find the maximum of the left-hand-side. However, that turns out to be about 11.72 for n = 3. (Presuming n is an integer.) Maybe I'm misinterpreting something -- I don't see where the "32" answer comes from, or your "17" for that matter.