Find the value range of $p$ if interval $(1,2)$ lies between the roots of $$2^x + 2^{2-x} + p = 0.$$
The answer is $(-\infty,-5)$.
My ''solution'':
Let $t= 2^x$, so $t \in(2,4)=:I$. Now you can write $$f(t) = -t-{4\over t}$$ Clearly since $t+{4\over t}\geq 4$ so $p\leq -4$. Since $f'(t)=-1+4t^{-2}$ we see that $f'(t)<0$ for $t\in I$ so $f$ is decreasing and thus $f_{\min} = f(4) =-5$. So given equation has a solution if $-5\leq p\leq-4$.
Clearly this is not correct answer since I try $p=-6$ which works.
I can't figure out what I'm doing wrong.
What you've determined are conditions for which your equation has a solution in the interval $(1, 2)$. But that isn't the problem you're trying to solve. You want to know when $(1, 2)$ is between the roots of your equation, i.e. when the equation has one root in $(-\infty, 1]$ and one root in $[2, +\infty)$.