Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.
A) 4.66
B) 3.14
C) 2.69
D) 4.66.

Question

Quizplus · Accepted Answer

Firstly, we notice that $x>0$ since the denominator $1+2^x$ is always positive. Plotting the graphs of $y=20/(1+2^x)$ and $y=x$, we can see that they intersect at one point in the interval $(0,5)$. To determine this intersection point, we can use numerical methods such as the bisection method or Newton-Raphson method. Using any of these methods, we can find that $x\approx2.69$. Thus, the answer is $\boxed{	extbf{(C)}}$.

Note: It is important to check that this is the only solution in the given interval by checking the concavity of the functions. In this case, we can show that $y=20/(1+2^x)$ is concave up while $y=x$ is concave down, which implies that they can intersect at most twice. Since we have found one intersection, we can conclude that there is only one solution in the given interval.

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Solve the Equation $\frac { 20 } { 1 + 2 ^ { x } } = x$