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Q.
If $f( x )$ is a twice differentiable function and given that $f (1)=1, f (2)=4, f (3)=9$, then
Application of Derivatives
Solution:
$f(1)=1, f(2)=4, f(3)=9 $
$\text { Let } g(x)=f(x)-x^2$
$\text { We have } g(1)=0, g(2)=0, g(3)=0$
hence by Rolles Theorem $g ^{\prime}( x )=0$ for some $c \in(1,2)$ are $g^{\prime}(x)=0$ for some $d \in(2,3)$ again using Rolles Theorem
for $g^{\prime}(x)=f^{\prime}(x)-2 x $
$f^{\prime \prime}(x)=0 \text { for some } x \in(c, d) $
$f^{\prime \prime}(x)-2=0 \text { for some } x \in(c, d) $
$f^{\prime \prime}(x)=2 \text { for some } x \in(1,3)$
