My books says to use calculus and algebra to prove this inequality, assuming that x is a positive number:
ln(1+x) less than or equal to x/(sqrt(1+x)).
I have no idea what to do and would greatly appreciate any help. Thanks in advance.
Consider: $\displaystyle \frac{1}{x+1}\leq{\frac{x+2}{2\cdot{(x+1)\cdot{\sq rt[]{x+1}}}}}$
Which clearly holds for $\displaystyle x\geq{0}$
Just to see: $\displaystyle \frac{1}{x+1}\leq{\frac{x+2}{2\cdot{(x+1)\cdot{\sq rt[]{x+1}}}}}$
If and only if (since $\displaystyle x\geq{0}$) : $\displaystyle 1\leq{\frac{x+2}{2\cdot{\sqrt[]{x+1}}}}$
Which is:
$\displaystyle 1\leq{\frac{\sqrt[]{x+1}}{2}+\frac{1}{2\cdot{\sqrt[]{x+1}}}}$
Let: $\displaystyle z=\sqrt[]{x+1}\geq{1}$
So: $\displaystyle 1\leq{\frac{z}{2}+\frac{1}{2\cdot{z}}}$
Multiplying by z: $\displaystyle z\leq{\frac{z^2}{2}+\frac{1}{2}}$ that is true since (z-1)^2>=0 and then zē+1>=2z
Integrating the inequality: $\displaystyle \int_0^b\frac{dx}{x+1}\leq{\int_0^b\frac{x+2}{2\cd ot{(x+1)\cdot{\sqrt[]{x+1}}}}}dx$ for $\displaystyle b\geq{0}$ What do you see?
Both expressions are equal when x=0, but their derivatives have the following property: $\displaystyle (\ln(x+1))'=\frac{1}{x+1}\leq{\frac{x+2}{2\cdot{(x +1)\cdot{\sqrt[]{x+1}}}}}=(\frac{x}{\sqrt[]{1+x}})'$
(as shown in the other post)
for $\displaystyle x\geq{0}$