First slide

Properties of ITF

Question

The greatest and least values of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ are

Moderate

A

$- \frac{π}{2}, \frac{π}{2}$
B

$- \frac{π^{3}}{8}, \frac{π^{3}}{8}$
C

$\frac{π^{3}}{32}, \frac{7 π^{3}}{8}$
D

none of these

Solution

We have,
   $\begin{array}{l} {(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3} \\ = {(\sin^{- 1} x + \cos^{- 1} x)}^{3} - 3 \sin^{- 1} x \cos^{- 1} x (\sin^{- 1} x + \cos^{- 1} x) \end{array}$
   $\begin{array}{l} = \frac{π^{3}}{8} - 3 (\sin^{- 1} x \cos^{- 1} x) \frac{π}{2} \\ = \frac{π^{3}}{8} - \frac{3 π}{2} (\frac{π}{2} - \sin^{- 1} x) \sin^{- 1} x \end{array}$
$= \frac{π^{3}}{8} - \frac{3 π^{2}}{4} \sin^{1} x + \frac{3 π}{2} {(\sin^{- 1} x)}^{2}$
   $\begin{array}{l} = \frac{π^{3}}{8} + \frac{3 π}{2} \{{(\sin^{- 1} x)}^{2} - \frac{π}{2} \sin^{- 1} x\} \\ = \frac{π^{3}}{8} + \frac{3 π}{2} \{{(\sin^{- 1} x - \frac{π}{4})}^{2}\} - \frac{3 π^{3}}{32} \\ = \frac{π^{3}}{32} + \frac{3 π}{2} {(\sin^{- 1} x - \frac{π}{4})}^{2} \end{array}$
So, the least value of ${(\sin^{- 1} x)}^{3} + {(\cos^{- 1} x)}^{3}$ is $\frac{π^{3}}{32}$
   ${(\sin^{- 1} x - \frac{π}{4})}^{2} \leq {(\frac{3 π}{4})}^{2}$
$∴$ Greatest value $= \frac{π^{3}}{32} + \frac{9 π^{2}}{16} \times \frac{3 π}{2} = \frac{7 π^{3}}{8}$

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