# how to do this ?

• Jun 13th 2006, 05:14 PM
yakkow
how to do this ?
Determine whether the function satisfies the hypotheses of the mean value theorem for the given interval

f(x) = tan^(-1) x , [-1,1]
• Jun 13th 2006, 05:23 PM
ThePerfectHacker
Quote:

Originally Posted by yakkow
Determine whether the function satisfies the hypotheses of the mean value theorem for the given interval

f(x) = tan^(-1) x , [-1,1]

Is is countinous on the closed interval [-1,1]--->Yes
Is is differenciable on the open interval (-1,1)---->Yes
Conditions satisfied.
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To find such a number you need.

$\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$
Since,
$\displaystyle f'(x)=\frac{1}{1+x^2}$
We have,
$\displaystyle f'(c)=\frac{1}{1+c^2}$
Also, $\displaystyle f(b)=f(1)=\pi/4$, $\displaystyle f(a)=f(-1)=-\pi/4$
Thus,
$\displaystyle \frac{1}{1+c^2}=\frac{\pi/4-(-\pi/4)}{1-(-1)}=\frac{\pi/2}{2}=\frac{\pi}{4}$
Thus,
$\displaystyle 1+c^2=\frac{4}{\pi}$
Thus,
$\displaystyle c^2=\frac{4-\pi}{\pi}$
Thus,
$\displaystyle c=\pm \sqrt{\frac{4-\pi}{\pi}$