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# Examine the applicability of Mean Value Theorem for all three functions given in $(iii)\;f (x) = x^2-1 \;for\; x \: \in [1,2]$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

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Toolbox:
• By Mean value theorem $f'(c)=\large\frac{f(b)-f(a)}{b-a}$
Step 1:
$f(x)=x^2-1$ for $x\in [1,2]$
It is polynomial.Therefore it is continuous in the interval [1,2].
$f'(x)=2x$
$f(1)=1-1=0$
$f(2)=4-1=3$
$f'(c)=2c$
Step 2:
$f'(c)=\large\frac{f(b)-f(a)}{b-a}$
$2c=\large\frac{3-0}{2-1}$
$2c=\large\frac{3}{1}$
$2c=3$
$c=\large\frac{3}{2}$
$c=\large\frac{3}{2}$ which belongs to $(1,2)$
answered May 14, 2013