How to find the maximum value of $3^x + 5^x - 9^x + 15^x - 25^x$ as $x$ varies over the reals? 
How to find the maximum value of  $3^x + 5^x - 9^x + 15^x - 25^x$ as x
  varies over the reals ?

Suggestions please!

Almost Same as grant_chat.
Let $a=3^x$ and $b=5^x\;,$ Then expression convert into $f(a,b) = a+b-a^2+ab-b^2$
So $$f(a,b) = -\frac{1}{2}[2a^2+2b^2-2ab-2a-2b]$$
So $$f(a,b) = -\frac{1}{2}\left[(a^2-2a+1)+(b^2-2b+1)+(a^2+b^2-2ab)-2\right]$$
So $$f(a,b) = -\frac{1}{2}\left[(a-1)^2+(b-1)^2+(a-b)^2-2\right]$$
So $$f(a,b) = \frac{1}{2}\left[2-(a-1)^2-(b-1)^2-(a-b)^2\right]\leq 1$$
And equality hold when 
$$(a-1)^2=0\Rightarrow a=1\Rightarrow 3^x=1=3^0$$ and $$(b-1)^2=0\Rightarrow b=1\Rightarrow 5^x=1=5^0$$ and $$(a-b)^2=0\Rightarrow a=b\Rightarrow 3^x=5^x$$
So from above we get $x=0$ for which $f(x)=3^x+5^x-4^x+15^x-9^x.$ is maximum.
And $\max{f(x)}=1$ at $x=0$ 
