"Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :"

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Accepted Answer

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$$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10\,\forall x \in [ - 1,1]$$

$$ \Rightarrow f'(x) = - {2 \over {\sqrt {1 - {x^2}} }} - {4 \over {1 + {x^2}}} - 6x - 2 < 0\,\forall x \in [ - 1,1]$$

So f(x) is decreasing function and range of f(x) is [f(1), f($-$1)], which is [$\pi$ + 5, 5$\pi$ + 9]

Now $4a - b = 4(\pi + 5) - (5\pi + 9)$

$= 11 - \pi$

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Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :

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Let $f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$, $x \in [ - 1,1]$. If [a, b] is the range of the function f, then 4a $-$ b is equal to :

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