The function form is $a\cdot 2^{bx} + c\cdot 2^{dx}$ where $a,b,c,d$ is some constant $>0$, and variable is $x$.

Question

For example, $0.2\cdot2^{x/2}+0.5\cdot2^{x/5}\leq C$.

In this case, the I want to find the variable $x$ to satisfy inequality constant $C$. Because the $2^{x/2}$ and $2^{x/5}$ have different denominator, so I can't not find the maximum value x which satisfy the constraint.

Any body help me? If anyone know the approximation form, please tell me. Actually, I need a upper bound of this form equation. $A\cdot 2^{x/B}+C\cdot 2^{x/D}$ where, $A,B,C,D$ is constant and positive value.

Using arithmetic-geometric inequality, I can find the lower bound of this form. However, I can not find the upper bound form.

I'm sorry, my english skill is bad.

Answer 1

For example, $0.2\cdot2^{x/2}+0.5\cdot2^{x/5}\leq C$.

In this case, the I want to find the variable $x$ to satisfy inequality constant $C$.

The LHS is a strictly increasing function in $x\,$, so the inequality will hold up to some value of $x$ where it becomes an equality, which value is the (unique) root of $0.2\cdot2^{x/2}+0.5\cdot2^{x/5} = C\,$.

Let $u=2^{x/10}$ then the latter equation becomes $0.2 \cdot u^5 + 0.5 \cdot u^2 - C = 0\,$. This is a quintic, which cannot be solved by radicals (in general), so you'll probably want to solve it numerically for $u\,$ then calculate $\,x = 10\,\log_{2} u\,$. (Or maybe solve the original equation numerically, directly.)

Actually, I need a upper bound of this form equation. $A\cdot 2^{x/B}+C\cdot 2^{x/D}$ where, $A,B,C,D$ is constant and positive value.

This is a very different question from the previous one, and the answer is negative: there exists no such upper bound, since $\,\lim_{x \to \infty} A\cdot 2^{x/B}+C\cdot 2^{x/D} = \infty\,$ when all constants are positive.