If the minimum value of fx-sin -1 x2-2 -cos -1 x-2 is a -2b where a- b are coprime- then the value of a-b is equal to

F(x)=(sin ^ 1 x)^2+2 πcos ^ 1 x+π^2 ⇒ f(x)=(sin ^ 1 x)^2+2 π(π2 sin ^ 1 x)+π^2 ⇒ f(x)=(sin ^ 1 x)^2 2 πsin ^ 1 x+π^2 +π^2 nbsp; f(x)=(sin ^ 1 x π)^2+π^2 nbs ...

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Standard XII

Mathematics

Greatest Integer Function

If the minimu...

Question

If the minimum value of $f (x) = {({sin}^{- 1} x)}^{2} + 2 π {cos}^{- 1} x + π^{2}$ is $\frac{a π^{2}}{b}$ where $a, b$ are coprime, then the value of $a + b$ is equal to

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Solution

$f (x) = {({sin}^{- 1} x)}^{2} + 2 π {cos}^{- 1} x + π^{2}$
$\Rightarrow f (x) = {({sin}^{- 1} x)}^{2} + 2 π (\frac{π}{2} - {sin}^{- 1} x) + π^{2}$
$\Rightarrow f (x) = {({sin}^{- 1} x)}^{2} - 2 π {sin}^{- 1} x + π^{2} + π^{2}$
$f (x) = {({sin}^{- 1} x - π)}^{2} + π^{2}$
$f_{min} = {(\frac{π}{2} - π)}^{2} + π^{2} = \frac{5 π^{2}}{4}$
$∴ a + b = 5 + 4 = 9$

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If the minimum value of f(x)=(sin−1x)2+2πcos−1x+π2 is aπ2b where a,b are coprime, then the value of a+b is equal to

f(x)=(sin−1x)2+2πcos−1x+π2 ⇒f(x)=(sin−1x)2+2π(π2−sin−1x)+π2 ⇒f(x)=(sin−1x)2−2πsin−1x+π2+π2 f(x)=(sin−1x−π)2+π2 fmin=(π2−π)2+π2=5π24 ∴a+b=5+4=9

If the minimum value of $f (x) = {({sin}^{- 1} x)}^{2} + 2 π {cos}^{- 1} x + π^{2}$ is $\frac{a π^{2}}{b}$ where $a, b$ are coprime, then the value of $a + b$ is equal to