Question

So, h(t) is twice-differentiable with these values:
When t=0, h(t)=1 and h'(t)=-9
t=1, h(t)=-4 and h'(t)=1
t=2, h(t)=-2 and h'(t)=3
t=3, h(t)=-1 and h'(t)=1
t=4, h(t)=-4 and h'(t)=-8


Justifying your reasoning, answer the following:
a) Determine where h(t) = 0 on the interval (0, 4).
b) Determine where h'(t) = 0 on the interval (0, 4).
c) Determine where h"(t) = 0 on the interval (0,4).

Answer 1

where as in "exactly for what value of t" or where as in "in what interval"?

Answer 2

for example, by the mean value theorem we know that there exists a c in the inteval (1,3) for which
\[h''(c)=\frac{h'(3)-h'(1)}{3-1}=\frac{1-1}{2}=0\] but that just tells us that h'' as a zero in the interval (1,3) not exactly where it is.
unless i miss my guess you cannot know exactly for what value of c is h''(c) = 0

Answer 3

maybe i am missing something

Answer 4

That's what I thought, I can find the intervals but not the exact value and I thought I was just going crazy. But I dont know what they mean to ask

Answer 5

really unless i am totally clueless i cannot imagine that it is asking for exact values

Answer 6

so i am going to assume the "where" means "on what interval in (0,4)" not
"for what number"

Answer 7

in which case it should be clear by mvt right?

Answer 8

Yeah. If its an interval, I think I got it. Thanks!