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Calculus Level 4

$f (x) = {\cos}^{-1} \sqrt{ {\sin}^{-1} \left[\sec \left( \ln \left(\dfrac{2x^2 + 3x -2}{x^2 - 3x + 2} \right)\right) \right]}$ Find the range of the above function.

Enter your answer as $( \Sigma {{\alpha}_{i}}^{2}) + 1$ where ${\alpha}_{i}$ represents the integers in the set of range.

If there are no integers in the range of $f (x)$ then enter your answer as 0.

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-2 10 2 4 0 -1 1 3

2 solutions

Prakhar Bindal
May 3, 2016

As usual a nice problem it is.

domain of inverse sine is from -1 to 1 both included.

Range of sec is less than equal to -1 or greater than equal to 1

Equality holds if output of sec is 1 or -1.

notice if input in inverse sin is 1 or -1 out will be pi/2 or -pi/2

input in cos is square root hence we should discard -pi/2 case.

input in cos is root (pi/2) which is greater than 1 but domain of inverse cos is again -1 to 1 both included.

Check that output of ln will be R as the given rational function takes all positive real values hence no problem there

Hence no x will satisfy this function and hence range is phi

Aniket Sanghi
May 1, 2016

The Range of F (x) is $\phi$

Hence no integers in the Range. ( quite unusual ! The end statement yields the answe!!! Right na)

Can you please post a solution. I am kinda confused why answer is $\phi$ and i couldn't calculate it precisely (though i got it correct :P)

neelesh vij - 5 years, 1 month ago

bro i am posting a solution

Prakhar Bindal - 5 years, 1 month ago

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