Question

the table gives selected values of the continuous function f.

x	-3	0	5	17

below is islas attempt to write a formal justification for the fact that the equation f(x) = 200 has a solution where 0 ≤ x ≤ 5.

is islas justification complete? if not, why?

islas justification:

we are given that f is continuous.

so, according to the intermediate value theorem, f(x) = 200 must have a solution when x is between x = 0 and x = 5.

choose 1 answer:

a) yes, islas justification is complete.

b) no, isla didnt establish that 200 is between f(0) and f(5).

c) no, isla shouldve mentioned that f is continuous.

Explanation:

The Intermediate Value Theorem (IVT) states that if a function \( f \) 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 \( c \) in \((a, b)\) such that \( f(c)=k \). For Isla's justification, she used IVT but didn't check if \( 200 \) lies between \( f(0) = 230 \) and \( f(5)=175 \). Since \( 175<200<230 \), \( 200 \) is between \( f(0) \) and \( f(5) \), but Isla didn't establish this step, which is required for IVT application. Option A is wrong as justification is incomplete. Option C is wrong as Isla did mention \( f \) is continuous. So the correct answer is B.

Answer:

B. No, Isla didn't establish that 200 is between \( f(0) \) and \( f(5) \).