Find the maximum value of $2x + 2\sqrt{x(1-x)}$ if $0 \leq x \leq 1.$ Find the maximum value of $2x + 2\sqrt{x(1-x)}$ when $0 \leq x \leq 1.$
I'm trying to find the upper bound of the expression, but I'm having trouble on how to start/how to solve the problem. Can someone help? Thanks in advance!
 A: Hint : Try to plug $x = cos^{2}(\theta)$ and use some trigonometry to see what can be the maximum value of $sin(A) + cos(A)$.
I hope this helps. If you are stuck, let me know!
Update : Since the OP seems not to follow the hint and work through it, I'll give the complete solution:
Plug in $x = cos^{2}(\theta)$, then we get:
$ 2x + 2\sqrt{x(1-x)} = 2cos^{2}(\theta) + 2 \sqrt{cos^{2}(\theta)(1-cos^{2}(\theta))} = 2cos^{2}(\theta) + 2 \sqrt{cos^{2}(\theta)sin^{2}(\theta)}$ $\mbox{ 
      } $ , using the identity $cos^{2}(\theta) + sin^{2}(\theta) = 1$
$ = 2cos^{2}(\theta) + 2 sin(\theta) cos(\theta) = [1 + cos(2\theta)] + sin(2\theta)$ $\mbox{  }$ (using the identities : $cos(2\theta) = 1 - 2cos^{2}(\theta)$ and $sin( 2\theta) = 2 sin(\theta) cos(\theta)$ )
Now we find maximum of $cos(2\theta) + sin(2\theta)$ :
$cos(2\theta) + sin(2\theta) = \sqrt{2} [1/\sqrt{2} sin(2\theta) + 1/\sqrt{2} cos(2\theta)] = \sqrt{2} [cos(\pi/4) sin(2\theta) + sin(\pi/4) cos(2\theta)] = \sqrt{2}sin( \pi/4 + 2\theta)$
using the identity $sin( A +B) = sin(A)cos(B) + cos(A)sin(B)$
Since maximum value of $sin( \pi/4 + 2\theta) = 1$, $cos(2\theta) + sin(2\theta) \leq \sqrt{2}$.
Thus, $ 2x + 2\sqrt{x(1-x)} \leq 1 + \sqrt{2}$.
