True or False?

i.) If f is defined on [2,10], f(2)=5, and f(10)=2, then there must be a number c in [2,10] where f(c)=4.

Answer: False because f is not necessarily continuous. Give a counterexample!










ii) If f is defined on [-6,-1] and u is between f(-6) and f(-1), then there must be a number c in [-6,-1] such that f(c)=u

Answer: False because f is not necessarily continuous. Give a counterexample! 












iii) If f is defined on [a,b] and w is between f(a) and f(b), then there must be some value c in [a,b] such that f(c)=w

Answer: False because f is not necessarily continuous. Give a counterexample!









iv) If f is continuous on [-2,0], f(-2)=3, and f(0)=0, then there must be at least one number c in [-2,0] where f(c)=4.5

Answer: False because 4.5 is not between f(-2)=3 and f(0)=0. Give a counterexample!













v) If f is continuous on (2,10), f(2)=-9, and f(10)=5, then there must be some number c in (2,10) where f(c)=3.

Answer: False because f is not continuous on [2,10]. Give a counterexample!









vi) If f is continuous on [2,10], f(2)=-9, and f(10)=5, then there must be some number c in (2,10) where f(c)=3.

True! By the Intermediate Value Theorem!








vii) If f is continuous on [2,4] and L is between f(2) and f(4), then there must be at least one point c in [2,4] with f(c)=L

True! By the Intermediate Value Theorem!







viii) If f is continuous on [-8,10] and L is between f(-8) and f(10), then there must be a value c in [-8,10] where f(c)=L

True! By the Intermediate Value Theorem!









ix) If f is continuous on [-7,0], f(-7)=-8, and f(0)=-5, then there must be at least one point c in [-7,0] such that f(c)=-9.

False because -9 is not between f(-7)=-8 and f(0)=-5. Give a counterexample!














x) If f is continuous on [-7,0], f(-7)=-8, and f(0)=-5, then there must be at least one point c in [-7,0] such that f(c)=-6.

True! By the Intermediate Value Theorem! Note that -6 is between f(-7)=-8 and f(0)=-5.