Post by Karl on Jul 26, 2015 23:30:22 GMT
On problem set 6, problem #4, I'm not sure if I'm interpreting parts (b) and (c) correctly.
Part (b) reads:
``If $f$ is a continuous real-valued function on $[a,b]$ and $\int_a^b f(s) ds = 0$, then $f \equiv 0$, then there exists a point $\theta \in [a,b]$ such that $f'(\theta) = 0$.''
Is the part "then $f \equiv 0$," supposed to be there? That is, are we supposed to show that then $f \equiv 0$ AND that there exists a point $\theta \in [a,b]$ such that $f'(\theta) = 0$.?
Finding the former makes the latter pretty boring.
Part (c) reads:
``If $f$ is a continuous real-valued function on $[a,b]$ then there exists a point $\theta \in [a,b]$ such that $\int_a^b f(s) ds = (b-a) f'(\theta)$.''
Thinking about areas under a graph, I think it might make more sense to as $f(\theta)$, rather than $f'(\theta)$.
Is that correct? If not, have others been successful in addressing these questions?
Cheers,
Karl
Part (b) reads:
``If $f$ is a continuous real-valued function on $[a,b]$ and $\int_a^b f(s) ds = 0$, then $f \equiv 0$, then there exists a point $\theta \in [a,b]$ such that $f'(\theta) = 0$.''
Is the part "then $f \equiv 0$," supposed to be there? That is, are we supposed to show that then $f \equiv 0$ AND that there exists a point $\theta \in [a,b]$ such that $f'(\theta) = 0$.?
Finding the former makes the latter pretty boring.
Part (c) reads:
``If $f$ is a continuous real-valued function on $[a,b]$ then there exists a point $\theta \in [a,b]$ such that $\int_a^b f(s) ds = (b-a) f'(\theta)$.''
Thinking about areas under a graph, I think it might make more sense to as $f(\theta)$, rather than $f'(\theta)$.
Is that correct? If not, have others been successful in addressing these questions?
Cheers,
Karl