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### $\sin ({\pi ^x}) = {e^x} + {e^{ - x}}$Solve for real x

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

$\frac{{{e^x} + {e^{ - x}}}}{2} \ge \sqrt {{e^x}.{e^{ - x}}}$

$\Rightarrow {e^x} + {e^{ - x}} \ge 2$

$\therefore \sin ({\pi ^x}) \ge 2$ which is not possible for any real x.

So, no solution.

### $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, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $\Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$