Question

a continuous function y = f(x) is known to be negative at x = 4 and positive at x = 6. why does the equation f(x)=0 have at least one solution between x = 4 and x = 6? illustrate with a sketch.
why does the equation f(x)=0 have at least one solution between x = 4 and x = 6?
a. f(x)=0 has at least one solution between x = 4 and x = 6 because f(x) must pass through all values between f(4) and f(6), regardless of whether f is continuous.
b. f(x)=0 has at least one solution between x = 4 and x = 6 because f is a continuous function on the closed interval 4, 6, and if y0 is any value between f(4) and f(6), then y0 = f(c) for some c in 4, 6.
c. f(x)=0 has at least one solution between x = 4 and x = 6 because all continuous functions have at least one zero over any nonempty closed interval.
choose a graph below that illustrates the situation.

Explanation:

The Intermediate - Value Theorem states that if a function $y = f(x)$ is continuous on a closed interval $[a,b]$, and $y_0$ is a value between $f(a)$ and $f(b)$, then there exists at least one $c\in[a,b]$ such that $y_0 = f(c)$. Here, $a = 4$, $b = 6$, $f(4)<0$ and $f(6)>0$, and $y_0 = 0$ is between $f(4)$ and $f(6)$. For the graph, we need a continuous curve that is below the $x$-axis at $x = 4$ and above the $x$-axis at $x = 6$, which means it must cross the $x$-axis (where $y=f(x)=0$) at some point in between.

Answer:

1. B. $f(x)=0$ has at least one solution between $x = 4$ and $x = 6$ because $f$ is a continuous function on the closed interval $[4,6]$, and if $y_0$ is any value between $f(4)$ and $f(6)$, then $y_0 = f(c)$ for some $c$ in $[4,6]$.
2. The graph that illustrates the situation is one where the curve representing $y = f(x)$ is below the $x$-axis at $x = 4$ and above the $x$-axis at $x = 6$ and is continuous. Without seeing the actual details of the graphs A, B, C, D, we know it should cross the $x$-axis between $x = 4$ and $x = 6$. If we assume a typical set - up where the $x$-axis is horizontal and $y$-axis is vertical, and the $x$-values range from $0$ to $10$ approximately, the correct graph would show a continuous curve going from negative $y$-values at $x = 4$ to positive $y$-values at $x = 6$.