$ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ How do I show that, for $ 0 < x < \dfrac{\pi}{4} $ (first quadrant), the inequality $ (1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 3x $ is valid?
I've tried Bernoulli's, but it took me to a false inequality (though all restrictions were respected). I actually thought of using Jensen's, but I don't know where to begin.

Use:
$$\cos x > 1/\sqrt{2},\quad \sin x > x - x^3/6>0$$
So that we have:
$$(1+\sin{x})^{\cos{x}} + (1+\cos{x})^{\sin{x}} > 
1+\left(1+\frac{1}{\sqrt{2}}\right)^{x-x^3/6}$$
Now, we can use the series expansion of $(1+a)^x$:
$$(1+a)^x = 1+ \log(1+a)x+\frac{1}{2}\log^2(1+a) x^2 + \ldots = 
\sum_{n=0}^\infty \frac{1}{n!}\log^n(1+a) x^n$$
Since all members of the series are positive for $a>0$, we can write:
$$1+\left(1+\frac{1}{\sqrt{2}}\right)^{x-x^3/6} > 
2+\log(1+1/\sqrt2)x>\frac{x}{2}+2>3x$$
With the last being true since $\pi/4<4/5$, the point where $3x=2+x/2$.
