Question

1. let ( f ) be a function with selected values given in the table above. which of the following statements must be true?

i. by the intermediate value theorem, there is a ( c ) in the interval ( (0,3) ) such that ( f(c) = 10 ).
ii. by the mean value theorem, there is a ( c ) in the interval ( (0,3) ) such that ( f(c) = -2 ).
iii. by the extreme value theorem, there is a ( c ) in the interval ( 0,3 ) such that ( f(c) leq f(x) ) for all ( x ) in the interval ( 0,3 ).
(a) i, ii, and iii
(o) i only
(i) ii only
(e) none

Explanation:

Step1: Check Intermediate Value Theorem (IVT) for I

IVT requires continuity on [0,3]. We only have discrete points, so no guarantee of continuity. Statement I is not necessarily true.

Step2: Check Mean Value Theorem (MVT) for II

MVT requires continuity on [0,3] and differentiability on (0,3). Discrete points don’t satisfy these; no guarantee of derivative existence. Statement II is not necessarily true.

Step3: Check Extreme Value Theorem (EVT) for III

EVT requires continuity on [0,3]. Discrete points lack continuity; no guarantee of a minimum point c. Statement III is not necessarily true.

Answer:

(E) None