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Find the least value of

${a^2}{\sec ^2}\theta + {b^2}{\rm{cose}}{{\rm{c}}^2}\theta $

The given expression can be written as,

$({a^2}{\sec ^2}\theta  - {a^2}) + {a^2} + ({b^2}{\rm{cose}}{{\rm{c}}^2}\theta  - {b^2}) + {b^2}$

$ = {a^2}{\tan ^2}\theta  + {b^2}{\cot ^2}\theta  + {a^2} + {b^2}$

Using A.M. $\geq $ G.M.,

$\frac{{{a^2}{{\tan }^2}\theta  + {b^2}{{\cot }^2}\theta }}{2} \ge \sqrt {{a^2}{{\tan }^2}\theta .{b^2}{{\cot }^2}\theta } $

$ \Rightarrow {a^2}{\tan ^2}\theta  + {b^2}{\cot ^2}\theta  \ge 2|ab|$

$ \Rightarrow {a^2}{\tan ^2}\theta  + {b^2}{\cot ^2}\theta  + {a^2} + {b^2} \ge 2|ab| + {a^2} + {b^2}$

$\therefore{({a^2}{\sec ^2}\theta  + {b^2}{\rm{cose}}{{\rm{c}}^2}\theta )_{\min }} = {a^2} + {b^2} + 2|ab|$

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$f(x)=x^6+2x^4+x^3+2x+3 $

$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$

$n=?$

Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R $. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$ is _ _ _ _ . Solution Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable. Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $ \Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$