Range of f(x,y)

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\hspace{-16}$If $\mathbf{x,y>0}$ and $\mathbf{f(x,y)=\sin^{-1}\left(\frac{x}{1+x^2}\right)+\sin^{-1}\left(\frac{y^2+y+1}{y^4+1}\right)}$\\\\\\ Then Find $\mathbf{\mathbb{R}}$ange of The Following $\mathbf{\mathbb{E}}$xpression.

#Mathematics #Calculus

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4 Answers

71

Vivek @ Born this Way ·2012-01-03 23:29:43

Is this : 0 â‰¤ f(x,y) â‰¤ 2 Î 3 ?

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71

Vivek @ Born this Way ·2012-01-06 01:05:35

man111, Please Confirm. If it's wrong, I'll try again ?

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1708

man111 singh ·2012-01-06 01:44:10

You are Right. vivek

Could you like to post your solution.

For Right Side is equal to sin is occuring or not (Not sure)

Thanks

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71

Vivek @ Born this Way ·2012-01-06 08:21:30

We have x,y as the independent variables, So we determine the Range of the two constituent expressions.

Now We see that,

0 < x1+x² â‰¤ 12 (x>0) and

0 < y²+y+1y⁴+1 â‰¤ 1 (Note that this function has values greater than 1 i.e, for instance take y = 0.5. But our domain Is restricted )

Putting in the given expressions,

0< sin^-1x1+x² â‰¤ Î 6

0 < sin^-1 y²+y+1y⁴+1 â‰¤ Î 2

Adding both We have,

0 < E â‰¤ 2Î 3 ; Where E is the given expression.

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Your Answer

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