Limit Continuity And Differentiability Ques 8
- Let $f$ be twice differentiable function satisfying $f(1)=1, f(2)=4, f(3)=9$, then
(2005, 2M)
(a) $f^{\prime \prime}(x)=2, \forall x \in(R)$
(b) $f^{\prime}(x)=5=f^{\prime \prime}(x)$, for some $x \in(1,3)$
(c) there exists atleast one $x \in(1,3)$ such that $f^{\prime \prime}(x)=2$
(d) None of the above
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Answer:
Correct Answer: 8.(c)
Solution: (c) Let, $g(x)=f(x)-x^2$
$\Rightarrow \quad g(x)$ has atleast $3$ real roots which are $x=1,2,3$
[by mean value theorem]
$\Rightarrow \quad g^{\prime}(x)$ has atleast $2$ real roots in $x \in(1,3)$
$\Rightarrow \quad g^{\prime \prime}(x)$ has atleast $1$ real roots in $x \in(1,3)$
$\Rightarrow \quad f^{\prime \prime}(x)-2 \cdot 1=0$. for atleast $1$ real root in $x \in(1,3)$
$\Rightarrow \quad f^{\prime \prime}(x)=2$, for atleast one root in $x \in(1,3)$
BETA
