First slide

Functions (XII)

Question

$If g : [- 2, 2] \to R, where f (x) = x^{3} + \tan x + [\frac{x^{2} + 1}{P}] is an odd function, then the value of parametric P, where [.]$ $d e n o t e s t h e g r e a t e s t i n t e g e r f u n c t i o n, i s$

Moderate

A

$- 5 < P < 5$
B

$P < 5$
C

$P > 5$
D

None of these

Solution

$\begin{array}{l} g (x) = x^{3} + \tan x + [\frac{x^{2} + 1}{P}] \\ or \begin{aligned} g (- x) = (- x)^{3} + \tan (- x) + [\frac{(- x)^{2} + 1}{P}] \\ = - x^{3} - \tan x + [\frac{x^{2} + 1}{P}] \end{aligned} \end{array}$
$\begin{aligned} Now, g (x) + g (- x) = 0 \\ Because g (x) is a odd function, \end{aligned}$
$\begin{array}{l} (x^{3} + \tan x + [\frac{x^{2} + 1}{P}]) + (- x^{3} - \tan x + [\frac{x^{2} + 1}{P}]) = 0 \\ or 2 [\frac{(x^{2} + 1)}{P}] = 0 or 0 \leq \frac{x^{2} + 1}{P} < 1 \\ Now, x \in [- 2, 2] \\ ∴ 0 \leq \frac{5}{P} < 1 or P > 5 \end{array}$

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