If f(x) is twice differentiable polynomial function such that f(1… - Vidyadip

MCQ

If $f(x)$ is twice differentiable polynomial function such that $f(1) = 1,f(2) = 4,f(3) = 9$, then

A
$f"(x) = 2,\forall x \in R$
✓
There exist at least one $x \in (1,\,3)$ such that $f"(x)=2 $
C
There exist at least one $x \in (2,\,3)$ such that $f'(x) = 5=f"(x)$
D
There exist at least one $x \in (1,\,2)$ such that $f(x) = 3$

✓

Answer

Correct option: B.

There exist at least one $x \in (1,\,3)$ such that $f"(x)=2 $

b
(b) Let a function be $g(x) = f(x) - {x^2}$

==> $g(x)$ has at least $3$ real roots which are $x = 1, 2 , 3$

==> $g'(x)$ has at least $2$ real roots in $x \in (1,\,3)$

==> $g"(x)$ has at least $1$ real roots in $x \in (1,\,3)$

==> $f'(x) = 2$ for at least one $x \in (1,3)$.

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