rule of differentiation: mean value theorem

**collegestudent321** · October 22nd 2008, 07:02 AM

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

**collegestudent321** · October 22nd 2008, 07:55 AM

anyone????

**Tony2710** · October 22nd 2008, 07:55 AM

Hi,

The first thing I'd do is get rid of the square root by squaring both sides of the equation. then cancel out some x's and units, and you should be just about there with the proof.

**collegestudent321** · October 22nd 2008, 07:59 AM

yea, i get how to do it algebraically, but it says to use the mean value theorem. that's what has me confused.

**Plato** · October 22nd 2008, 08:06 AM

Suppose that $x > 0$ . Because $f(x) = \sqrt {1 + x}$ has a derivative, by the mean value theorem
$\left( {\exists c \in \left( {0,x} \right)} \right)\left[ {\frac{1}{{2\sqrt {1 + c} }} = \frac{{\sqrt {1 + x} - 1}}{x}} \right]$ .
But we know that $\frac{1}{{2\sqrt {1 + c} }} < \frac{1}{2}$ .
Now you put all that together to finish it all off.

**collegestudent321** · October 22nd 2008, 08:14 AM

wow, ok, i see. thank you soooo much!

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