Question

the table gives selected values of the continuous function f.
x 0 2 5 8
f(x) 0 100 40 120
below is kristins attempt to write a formal justification for the fact that the equation f(x) = 80 has a solution where 2 ≤ x ≤ 5.
is kristins justification complete? if not, why?
kristins justification:
f(2) = 100 and f(5) = 40, so 80 is between f(2) and f(5).
so, according to the intermediate value theorem, f(x) = 80 must have a solution somewhere between x = 2 and x = 5.
choose 1 answer:
a yes, kristins justification is complete.
b no, kristin didnt establish that 80 is between f(2) and f(5).
c no, kristin 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 \( f(c)=k \). Kristin used IVT but didn't mention the continuity of \( f \), which is a necessary condition for IVT to apply. Option A is wrong as the justification is incomplete. Option B is wrong because \( 80 \) is between \( 40 \) (\( f(5) \)) and \( 100 \) (\( f(2) \)), so she did establish that. Option C is correct as continuity of \( f \) on \([2, 5]\) is required for IVT.

Answer:

C. No, Kristin should've mentioned that \( f \) is continuous.