hi, guys. i'm having trouble solving this problem. i have a feeling i am making it more difficult than it really is. I really need help because the hw is due today! please help!
show that (1+x)^(1/2) < 1 + (1/2)x if x > 0
hi, guys. i'm having trouble solving this problem. i have a feeling i am making it more difficult than it really is. I really need help because the hw is due today! please help!
show that (1+x)^(1/2) < 1 + (1/2)x if x > 0
Suppose that $\displaystyle x > 0$. Because $\displaystyle f(x) = \sqrt {1 + x} $ has a derivative, by the mean value theorem
$\displaystyle \left( {\exists c \in \left( {0,x} \right)} \right)\left[ {\frac{1}{{2\sqrt {1 + c} }} = \frac{{\sqrt {1 + x} - 1}}{x}} \right]$.
But we know that $\displaystyle \frac{1}{{2\sqrt {1 + c} }} < \frac{1}{2}$.
Now you put all that together to finish it all off.