Problem 11

Explanation

We will simplify the form of $P$ first. Recall that $4^x = (2^2)^x = (2^x)^2$. Thus, the form $4^x - 1$ is a difference of squares which can be factored into $(2^x - 1)(2^x + 1)$.

Next, let's look at the form of $Q$:

We can see the relationship between $P$ and $Q$ by substituting $P = 2^x + 1$ into the equation for $Q$:

Given $0 < x < 1$. Let's analyze the value of $2^x$ in that range:

• If $x > 0$, then $2^x > 2^0 \implies 2^x > 1$.
• If $x < 1$, then $2^x < 2^1 \implies 2^x < 2$.

Thus, $1 < 2^x < 2$.

Back to the equation $P = Q \cdot 2^x$: Since $2^x > 1$ and $Q$ is positive (because $2^x > 0$), multiplying $Q$ by a number greater than $1$ will result in a value greater than $Q$ itself.

Therefore, the correct relationship is $P > Q$.