Finding the max of a function using $f(x)=L(x)-E(x)$ where $L(x)$ is linear and $E(x)$ is exponential

Question

Take this problem, which was on a younger friend's recent AP Precalc test:

_{(note that this test was given by an online provider called "AP Classroom". I highly doubt that this was tested beforehand)}

20. In the $xy$-plane, the graphs of a linear function $L$ and the exponential function $E$ both pass through the points $(0,2)$ and $(1,6)$. The function $f$ is given by $f(x)=L(x)-E(x)$. What is the maximum value of $f$?

Here is my friend's attempt at solving this:

1. Tackling $L(x)$

This is really easy to do. It can be noticed that due to rise/run, and since linear functions are defined by $mx+b$, we can see that it is intentional that $L(x)=4x+2$

2. Tackling $E(x)$

This is where there are multiple ways to represent the function, because apparently it wasn't mentioned whether this had to be in the form $ab^{x+c}+d$ or $ax^b+c$. My friend chose $E(x)=5^x+1$ because it satisfied both of the boundaries, and got that the max. of $f(x)$ was $\approx0.777$. (found through Desmos since it's allowed to be used on these tests)

Here are the options, which I intentionally didn't show beforehand:$$\begin{align}(a)&\quad0.007\\(b)&\quad0.172\\(c)&\quad0.54\\(d)&\quad1.002\end{align}$$Yep. Those are the only options. As you might have noticed, $0.777$ isn't on there. So, my friend randomly guessed (d), which turned out to be incorrect, with the correct answer apparently being (b). So my question is:

How is (b) 0.172 the correct answer?

Answer 1

The question is ill-defined. Exponential functions have three degrees of freedom. Any exponential function can be written as $$E(x) = a(b)^x + c.$$ Note that $$ab^{x+c} + d = ab^xb^c +d = \tilde{a}b^x + d,$$ where $\tilde{a} = a\cdot b^c$. Thus your first description of an exponential does not need the $+c$ in the exponent. (Further, your second description of an exponential is written as a polynomial, not an exponential. Perhaps there's a typo?)

Therefore, in order to get a unique description of a polynomial, at least 3 points would need to be provided.