The least value of $4se{c}^{2}x + 25 \cos e{c}^{2}x$ is ___ and is obtained at x =

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=4 {\sec }^2\mathrm{x}+25 {\mathrm{cosec}}^2\mathrm{x} \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=4\left(2 \sec \mathrm{x}\right)\sec \mathrm{x} \tan \mathrm{x}+25\left(2 \mathrm{cosec} \mathrm{x}\right)\left(-\mathrm{cosex} \mathrm{x}\right) \end{gathered}$$

$\mathrm{f}\left(\mathrm{x}\right)$ is min or max

$$\begin{gathered} \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=0 \\ \Rightarrow 8 {\sec }^2\mathrm{x} \tan \mathrm{x}=50 {\mathrm{cosec}}^2\mathrm{x} \cot \mathrm{x} \\ \Rightarrow \frac{8 \sin \mathrm{x}}{{\cos }^3\mathrm{x}}=50 \frac{\cos \mathrm{x}}{{\sin }^3\mathrm{x}} \\ \Rightarrow \frac{3{\sin }^4\mathrm{x}}{{\cos }^4\mathrm{x}}=\frac{50}{8}=\frac{25}{4} \\ \Rightarrow {\tan }^4\mathrm{x}={\left(5/2\right)}^2 \\ \Rightarrow \tan \mathrm{x}=\sqrt{5/2} \\ \Rightarrow \mathrm{x}={\tan }^{-1}\left(\sqrt{5/2}\right) \end{gathered}$$

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=4{\left(\frac{\sqrt{7}}{\sqrt{2}}\right)}^2+25{\left(\frac{\sqrt{7}}{\sqrt{5}}\right)}^2 \\ =14+35=49 \end{gathered}$$