First slide

Monotonicity

Question

A function s(x) is defined as $g (x) = \frac{1}{4} f (2 x^{2} - 1) + \frac{1}{2} f (1 - x^{2}) and f^{'} (x)$ is an increasing function. Then g(x) is increasing in the interval

Moderate

A

(-1, 1)
B

$(- \sqrt{\frac{2}{3}}, 0) \cup (\sqrt{\frac{2}{3}}, \infty)$
C

$(- \sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}})$
D

none of these

Solution

$g^{'} (x) = {xf}^{'} (2 x^{2} - 1) - {xf}^{'} (1 - x^{2}) = x (f (2 x^{2} - 1) - f (1 - x^{2})) \Rightarrow g^{'} (x) > 0$
If $x > 0, 2 x^{2} - 1 > 1 - x^{2}$ (as f ' is an increasing function)
or $3 x^{2} > 2 or x \in (- \infty, - \sqrt{\frac{2}{3}}) \cup (\sqrt{\frac{2}{3}}, \infty)$
or $x \in (\sqrt{\frac{2}{3}}, \infty)$
If $x < 0, 2 x^{2} - 1 < 1 - x^{2}$
or $3 x^{2} < 2 or x \in (- \sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}}) or x \in (- \sqrt{\frac{2}{3}}, 0)$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App