For the given function and interval, determine if we're allowed to use the Mean Value Theorem for the function on that interval. If so, what does the Mean Value Theorem let us conclude?
f(x) = x3on (-1, 1)
Answer
Since f is continuous on [-1,1] and differentiable on (-1,1), yes, we are allowed to use the Mean Value Theorem. The Mean Value Theorem says there will be some c in (-1,1) where
Example 2
For the given function and interval, determine if we're allowed to use the Mean Value Theorem for the function on that interval. If so, what does the Mean Value Theorem let us conclude?
on (1, 2)
Answer
Although f is discontinuous at 0, f is continuous on the interval [1, 2]. f is also differentiable on (1, 2), so we're allowed to use the Mean Value Theorem. The Mean Value Theorem tells us there will be some c in (1, 2) with
Example 3
For the given function and interval, determine if we're allowed to use the Mean Value Theorem for the function on that interval. If so, what does the Mean Value Theorem let us conclude?
on (0, 4)
Answer
f is continuous on [0,4] and differentiable on (0, 4) (but not at x = 0), so we are allowed to use the Mean Value Theorem here.
We can conclude that there is a c in (0, 4) with
Example 4
For the given function and interval, determine if we're allowed to use the Mean Value Theorem for the function on that interval. If so, what does the Mean Value Theorem let us conclude?
f(x) = 0 on (2, 3)
Answer
f is continuous everywhere, differentiable everywhere (with derivative equal to zero), and boring. We're allowed to use the Mean Value Theorem, and it tells us nothing exciting:for some c in (2,3),
on (0, 4)
