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Establishing differentiability for MVT

A function must be differentiable for the mean value theorem to apply. Learn why this is so, and how to make sure the theorem can be applied in the context of a problem.
The mean value theorem (MVT) is an existence theorem similar the intermediate and extreme value theorems (IVT and EVT). Our goal is to understand the mean value theorem and know how to apply it.

MVT and its conditions

The mean value theorem guarantees, for a function f that's differentiable over an interval from a to b, that there exists a number c on that interval such that f(c) is equal to the function's average rate of change over the interval.
Graphically, the theorem guarantees that an arc between two endpoints has a point at which the tangent to the arc is parallel to the secant through its endpoints.
The precise conditions under which MVT applies are that f is differentiable over the open interval (a,b) and continuous over the closed interval [a,b]. Since differentiability implies continuity, we can also describe the condition as being differentiable over (a,b) and continuous at x=a and x=b.
Using parameters like a and b and talking about open and closed intervals is important if we want to be mathematically precise, but these conditions essentially mean this:
For MVT to apply, the function must be differentiable over the relevant interval, and continuous at the interval's edges.

Why differentiability over the interval is important.

To understand why this condition is important, consider function f. The function has a sharp turn between x=a and x=b, so it's not differentiable over (a,b).
Indeed, the function has only two possible tangent lines, neither of which is parallel to the secant between x=a and x=b.

Why continuity at the edges is important.

To understand this, consider function g.
As long as g is differentiable over (a,b) and continuous at x=a and x=b, MVT applies.
Now let's change g so it's not continuous at x=b. In other words, the one-sided limit limxbg(x) remains the same, but the function value changes to something else.
Notice how all of the possible tangent lines on the interval are necessarily increasing, while the secant line is decreasing. So there isn't any tangent line that's parallel to the secant line.
In general, if a function isn't continuous at the edges, the secant line will be disconnected from the tangent lines along the interval.
In Problem set 1 we will analyze the applicability of the mean value theorem to function h at different intervals.
Problem 1.A
Does MVT apply to h over the interval [5,1]?
Choose 1 answer:

Problem 2
The graph of function f has a vertical tangent at x=2.
Does MVT apply to f over the interval [1,5]?
Choose 1 answer:

Want more practice? Try this exercise.
Notice: When MVT doesn't apply, all we can tell is that we aren't certain the conclusion is true. It does not mean that the conclusion isn't true.
In other words, it's possible to have a point where the tangent is parallel to the secant, even when MVT doesn't apply. We just can't be certain about it unless the conditions for MVT have been met.
For example, in the last problem, MVT didn't apply to f over the interval [1,5], even though there are actually two points in the interval [1,5] where the tangent is parallel to the secant between the endpoints.
Problem 3
This table gives a few values of function h.
James said that since h(7)h(3)73=1 there must be a number c in the interval [3,7] for which h(c)=1.
Which condition makes James's claim true?
Choose 1 answer:

Want more practice? Try this exercise.

Common mistake: Not recognizing when the conditions are met

Let's take Problem 3 for example. These are the common ways we would expect the conditions for MVT to look:
  • h is differentiable over (3,7) and continuous over [3,7].
  • h is differentiable over (3,7) and continuous at x=3 and x=7.
However, we will not always be given information about the function this way. For example, if h is differentiable over [3,7], the conditions are met because differentiability implies continuity.
Another example is when h is differentiable over a larger interval, for example (2,8). Even though continuity isn't mentioned, differentiability over (2,8) implies differentiability over (3,7) and continuity over [3,7].
Problem 4
f is a differentiable function. f(1)=2 and f(5)=2.
Match each conclusion with its appropriate existence theorem.

Common mistake: Applying the wrong existence theorem

By now, we are familiar with three different existence theorems: the intermediate value theorem (IVT), the extreme value theorem (EVT), and the mean value theorem (MVT). They have a similar structure but they apply under different conditions and guarantee different kinds of points.
  • IVT guarantees a point where the function has a certain value between two given values.
  • EVT guarantees a point where the function obtains a maximum or a minimum value.
  • MVT guarantees a point where the derivative has a certain value.
Before applying one of the existence theorems, make sure you understand the problem well enough to tell which theorem should be applied.

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