The maximum value of a trigonometric expression (without derivative) For how trivial it may appear, here I am with this question: find the value of $x$ for which this expression has its maximum.
$$f(x) = 1 + 4\sin^2(x) + 4\sin(x)\cos(x)$$
Without making use of derivatives, and only basing on trigonometric expressions / formulas. 
What I have tried is to manipulate a bit the expression, but I couldn't be able to reach any interesting expression.

Basics:
$$
\begin{align}
%
 \sin^{2} x &= \frac{1}{2} \left( 1 - \cos 2 x \right) \\[3pt]
%
 \cos x \sin x &= \frac{1}{2} \sin 2x
%
\end{align}
$$

The target function becomes
$$
  f(x)
  = 1 + 4 \sin^{2} x + 4 \sin x \cos x
  = 3 + 2 \left( \sin 2 x - \cos 2 x \right)
\tag{1}
$$

When does $\sin \theta - \cos \theta$ achieve maximal value? When
$$
 \theta = \frac{3}{4}\pi + 2\pi k, \quad k \in \mathbb{Z}
$$

Therefore, the function (1) achieves maximum value at
$$
 \boxed{x = \frac{3}{8}\pi + \pi k, \quad k \in \mathbb{Z}}
$$

