Question

Q1) Show that if x is positive, then

Ln(1+1/x) > 1/(1+x)

Q2.) Find the length of the curve y^2=x^3 from the origin to the point where the tangent makes an angle of 45 degrees with the x- axis.

Answer 1

\[f(x)=\ln (1+\frac{1}{x})-\frac{1}{1+x}\]\[f'(x)=\frac{-1/x^2}{1+\frac{1}{x}}+\frac{1}{(1+x)^2}=\frac{-1/x}{1+x}+\frac{1}{(1+x)^2}=\frac{1-\frac{1}{x}(1+x)}{(1+x)^2}=-\frac{1}{x(1+x)^2}\]decreasing from \(f(0)=+\infty\) to \(f(\infty)=0\) so never goes down the x-axis