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Rationalization of an Irrational Number

lf fx=A2-1/...

Question

lf $f (x) = (\frac{A^{2} - 1}{A^{2} + 1}) x^{3} - 3 x + e^{\frac{π}{2}}$ is a decreasing function of $x$ in $R$ then the set of possible value of $A$ which does not depends on x

[1, \infty)

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[- 1, 1]

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(- \infty, \infty)

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(- \infty, - 1)

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Solution

The correct option is C $[- 1, 1]$
$f^{'} (x) = \frac{3 (A^{2} - 1)}{(A^{2} + 1)} x^{2} - 3$
$= 3 (\frac{(A^{2} - 1)}{A^{2} + 1} x^{2} - 1)$
$f^{'} (x) < 0$
$\frac{(A^{2} - 1) x^{2}}{A^{+} 1} - 1 < 0$
$\frac{A^{2} - 1}{A^{2} + 1} \leq 0$
$A \in [- 1, 1]$

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