A continuous and differentiable function ‘ f ‘ satisfies the cond… - Vidyadip

MCQ

A continuous and differentiable function $‘ f ‘$ satisfies the condition ,$\int\limits_0^x \, f (t) d t = f^2 (x) - 1$ for all real $‘ x ‘$. Then :

A
$‘ f ‘$ is monotonic increasing $\forall x \in R$
B
$‘ f ‘$ is monotonic decreasing $\forall x \in R$
C
the graph of $y = f (x)$ is a straight line.
✓
both $(A)$ and $(C)$

✓

Answer

Correct option: D.

both $(A)$ and $(C)$

d
Differentiating $f (x) = 2 f (x). f ‘ (x) $

$\Rightarrow f ‘ (x) =$ $\frac{1}{2}$ $( f (x)) \ne 0)$
Hence $f (x) =$ $\frac{x}{2}$ $+ c$.

Put $x = 0$ ; $f (0) = c $; but $f^2 (0) = 1$
$\Rightarrow f (0) = ± 1$
Hence $f (x) = \frac{x}{2} ± 1$

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