Question

a continuous function y = f(x) is known to be negative at x = 8 and positive at x = 9. why does the equation f(x)=0 have at least one solution between x = 8 and x = 9? illustrate with a sketch.

why does the equation f(x)=0 have at least one solution between x = 8 and x = 9?

a. f(x)=0 has at least one solution between x = 8 and x = 9 because f is a continuous function on the closed interval 8,9, and if y0 is any value between f(8) and f(9), then y0 = f(c) for some c in 8,9.
b. f(x)=0 has at least one solution between x = 8 and x = 9 because all continuous functions have at least one zero over any non - empty closed interval.
c. f(x)=0 has at least one solution between x = 8 and x = 9 because f(x) must pass through all values between f(8) and f(9), regardless of whether f is continuous.

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 \(k\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one number \(c\) in the interval \((a,b)\) such that \(f(c)=k\). Here, \(a = 8\), \(b = 9\), \(f(8)<0\), \(f(9)>0\), and \(k = 0\). Since \(0\) is between \(f(8)\) and \(f(9)\) and \(f(x)\) is continuous on \([8,9]\), there must be at least one \(c\in(8,9)\) such that \(f(c)=0\).

Answer:

A. \(f(x)=0\) has at least one solution between \(x = 8\) and \(x = 9\) because \(f\) is a continuous function on the closed interval \([8,9]\), and if \(y_0\) is any value between \(f(8)\) and \(f(9)\), then \(y_0=f(c)\) for some \(c\) in \([8,9]\)