Is ln(1+x) Always Greater Than or Equal to x - x^2/2 for x>=0?

[peripatein]

Hello,

Homework Statement

I was asked to prove that for every x>=0, ln(1+x)>=(x-x^2/2).

Homework Equations

The Attempt at a Solution

I defined f(x) thus: f(x) = ln(1+x) + x^2/2 - x and found f'(x)=x^2/(1+x). I hence wrote that since f'(x) is always non-negative for every x>=0 (since x^2>=0 in that domain) f(x) is likewise always positive.
Does that suffice?

 

[Dick]

Hello,

Homework Statement

I was asked to prove that for every x>=0, ln(1+x)>=(x-x^2/2).

Homework Equations

The Attempt at a Solution

I defined f(x) thus: f(x) = ln(1+x) + x^2/2 - x and found f'(x)=x^2/(1+x). I hence wrote that since f'(x) is always non-negative for every x>=0 (since x^2>=0 in that domain) f(x) is likewise always positive.
Does that suffice?

Pretty much. You'll want to add a comment that f(0) is non-negative as well.