suppose that $f(x)$ is a continuous function with the following values:\n|$x$|0|1|2|3|4|5|\n|$f(x)$|2|5|10|8|…
suppose that $f(x)$ is a continuous function with the following values:\n|$x$|0|1|2|3|4|5|\n|$f(x)$|2|5|10|8|9|-4|\nwhat is the minimum number of solutions to $f(x)=0$ in the interval $0leq xleq5$\nthere are at least solutions.
Answer
Explanation:
Step1: Apply Intermediate - Value Theorem
The Intermediate - Value Theorem states that if (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). We want to find where (f(x) = 0).
Step2: Check sign - changes
We have (f(0)=2) and (f(5)= - 4). Since (f(x)) is continuous on ([0,5]) and (0) is between (f(0) = 2) and (f(5)=-4), by the Intermediate - Value Theorem, there must be at least one value (c\in(0,5)) such that (f(c) = 0).
Answer:
1