Let $x$ be a positive real number. Inequality problem with $\pi$ and $x$ terms.

Let $x$ be a positive real number. Then

(A) $ x^{2} + \pi^{2} + x^{2\pi}> x\pi+ (\pi+x)x^{\pi} $

(B) $ x^{\pi} + \pi^{x}> x^{2\pi}+ \pi^{2x} $

(C) $ \pi x +(\pi+x)x^{\pi}>x^2 + \pi^2 + x^{2\pi} $

(D) none of the above.

I've tried using AM GM inequality. But not been able to solve. It became more cumbersome. And I know it can be solved using substitution (x=1). But I want to know the process, intuition to such problems Help??

Solution 1:

To show something is not true, a counter example is enough. Substituting $x=1$ shows (B) and (C) are incorrect. What is left is to check (A). For this, some creative AM-GM works, add the following three AM-GMs:

$$\frac{x^2}2+\frac{\pi^2}2 \ge \pi x \tag{1}$$ $$\frac{x^2}2+\frac{x^{2\pi}}2 \ge x\cdot x^\pi \tag{2}$$ $$\frac{\pi^2}2+\frac{x^{2\pi}}2 \ge \pi\cdot x^\pi \tag{3}$$

Equality is possible iff $x=\pi = x^\pi$ which is never, so the inequality is strict. Hence (A) is correct.