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Theorems for Differentiability

12. if x+1/x ...

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12. if x+1/x = 2 and [sin-1y]= -2 then the minimum value of sin-1x + sin-1y is 1. pi/4 2. 0 3. -pi 4. 3pi/2

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Q. Assertion :If

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

\frac{3 \sum_{r = 1}^{2008} (x^{r} + y^{r})}{2 \sum_{r = 1}^{2008} (x^{r} y^{r})} = 3

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

is possible only if

x = y = z = 1

{sin}^{- 1} x + {sin}^{- 1} y = \frac{2 π}{3}

{cos}^{- 1} x - {cos}^{- 1} y = \frac{π}{3}

x \in (0, \frac{1}{2})

, then the number of value(s) of

y

satisfying the equation is

\frac{s i n^{2} 2 x + 4 s i n^{4} x - 4 s i n^{2} x c o s^{2} x}{4 - s i n^{2} 2 x - 4 s i n^{2} x} = \frac{1}{9}

π

, then value of x is-

{tan}^{- 1} (x^{2} + 3 | x | - 4) + {cot}^{- 1} (4 π + {sin}^{- 1} (sin 14)) = \frac{π}{2}

, then the value of

{sin}^{- 1} (sin 2 | x |)

{sin}^{- 1} x + {sin}^{- 1} y + {sin}^{- 1} z = \frac{3 π}{2}

, then find the minimum value of

x^{2} + y^{2} + z^{2}

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