<s>[INST] Number of Real Solutions of $\frac{7^{1+\cos(\pi x)}}{3}+3^{x^2-2}+9^{\frac{1}{2}-|x|}=1$ 
Find the number of real solutions of the equation $$\frac{7^{1+\cos(\pi x)}}{3}+3^{x^2-2}+9^{\frac{1}{2}-|x|}=1\,.$$

By hit and trial i got the solution at $x=\pm 1$ but i am not able to solve it as it involves power of 7
 [/INST] Note that $1+\cos(\pi x)\geq 0$ for all $x\in\mathbb{R}$.  Therefore,
$$1-\frac{7^{1+\cos(\pi x)}}{3}\leq 1-\frac{1}{3}=\frac{2}{3}\,.$$
Thus, if $x$ is a real solution of the required equation, then
$$3^{x^2-1}+3^{2\big(1-|x|\big)}=3\,\left(3^{x^2-2}+9^{\frac12-|x|}\right)=3\,\left(1-\frac{7^{1+\cos(\pi x)}}{3}\right)\leq 2\,.$$
By the AM-GM Inequality,
$$3^{x^2-1}+3^{2\big(1-|x|\big)}\geq 2\,\sqrt{3^{x^2-1}\cdot3^{2\big(1-|x|\big)}}=2\cdot3^{\frac{\big(|x|-1\big)^2}{2}}\geq 2\,.$$
Therefore, the inequality above must be an equality, which means $x^2-1=2\big(1-|x|\big)$ and $\big(|x|-1\big)^2=0$.  This shows that $|x|=1$, or $x=\pm1$.
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