Saturday, September 19, 2015

calculus - Prove with the mean value theorem that xfracx22<ln(1+x)




Prove with the mean value theorem that $x-\frac{x^2}{2}<\ln(1+x)(0,)



Approach
f(x):=ln(1+x) with the mean value theorem in [0,x]



11+ξ=ln(1+x)0x0




11+ξ takes the biggest value when ξ is 0



and so 1<ln(1+x)x multiply with x and you get
x<ln(1+x)



I can't prove the second part.


Answer



For MVT



ln(1+x)+x22(ln1+0)=x(11+c+c)>xc(0,x)




Indeed



11+c+c=c2+c+11+c>1


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...