Question

If f is a differentiable function and f(0)= -3 and f(6)= 6, then which of the following must be true?
i. There exists a c in [0,6] where f(c)=0
ii. There exists a c in [0,6] where f'(c)=0
iii. There exists a c in [0,6] where f'(c)= 3/2
a) ii only
b) i only
c) i and iii only
d) ii and iii only
e) i, ii, and iii

Answer 1

i'd say c

Answer 2

why would you say c?

Answer 3

hmm. first of all the choice has to include i. because f is continuous so it has to cross the x axis.

secondly, if f is monotonically increasing on [0,6] then there is no c in [0, 6] such that f'(c)=0. so this eliminates any choice that has ii.

Answer 4

so it comes down to only two choices... b or c.
in iii. , this looks like the mean value theorem...

Answer 5

what's the answer?