Multiple choice

The number of roots of the equation $1+3^{x/2}=2^{ x}$ is

  1. $0$
  2. $1$
  3. $2$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let f(x) = 1 + 3^(x/2) - 2^x. By testing values, f(0) = 1 + 1 - 1 = 1, f(2) = 1 + 3 - 4 = 0, and f(4) = 1 + 9 - 16 = -6. Since the function is continuous and changes sign, there is at least one root; checking the derivatives shows it is monotonic for x > 2, confirming exactly one root.

AI explanation

By substituting values for $x$, we get $1 + 3^0 = 2^0$ which yields $2 = 1$ (false) and $1 + 3^2 = 2^4$ which yields $10 = 16$ (false), but at $x = 2$ we have $1 + 3^1 = 2^2$ which yields $4 = 4$ (true). Analyzing the functions $f(x) = 2^x - 3^{x/2}$ reveals that its derivative changes sign only once as the terms exchange growth dominance. Because the continuous function $f(x)$ goes from positive to negative and eventually back to positive, it crosses zero exactly once. The number of real roots is 1.

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