If x∈−200,−50 and f(x)=tan⁡50∘+x+cos⁡50∘+x+cot⁡50∘+x then global maximum value f(x) is k, then the value of [k2 ] _________ (where [.] denotes greatest integer function).

tan⁡θ+cot⁡θ+cos⁡θ is decreasing function for θ∈300,450⇒maxima occurs atθ=300|a−b≤|a−b|| for a>0, b>0⇒k=12112⇒k2=12112

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If $x \in [- 20^{0}, - 5^{0}]$ and $f (x) = \tan (50^{\circ} + x) + \cos (50^{\circ} + x) + \cot (50^{\circ} + x)$ then global maximum value f(x) is k, then the value of [k² ] _________
(where [.] denotes greatest integer function).

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Detailed Solution

$\tan θ + \cot θ + \cos θ$ is decreasing function for $θ \in [30^{0}, 45^{0}]$
$\Rightarrow$ maxima occurs at
$\begin{array}{l} θ = 30^{0} | \sqrt{a} - \sqrt{b} \leq \sqrt{| a - b |} | \\ for a > 0, b > 0 \\ \Rightarrow k = \sqrt{\frac{121}{12}} \Rightarrow k^{2} = \frac{121}{12} \end{array}$