Video: Determining Which of a Set of Conditions Are Satisfied for a Function Differentiable at a Specific Point

If a function 𝑓 is differentiable at π‘₯ = 2, which of the following must be true? [A] lim_(π‘₯ β†’ 2) of 𝑓(π‘₯) exists. [B] 𝑓(2) exists. [C] 𝑓’(2) exists.

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Video Transcript

If a function 𝑓 is differentiable at π‘₯ is equal to two, which of the following must be true? I) The limit as π‘₯ approaches two of 𝑓 of π‘₯ exists. II) 𝑓 of two exists. And III, 𝑓 prime of two exists.

We’re told that our function 𝑓 is differentiable at π‘₯ is equal two. And there are a couple of things we know about the conditions that need to be satisfied for a function to be differentiable at a point. We’ll look at these conditions for our function and compare with the given options I, II, and III. The first thing we can say about a differentiable function is 𝑓 of π‘₯ is differential at the point π‘₯ equal to π‘Ž, if 𝑓 prime of π‘Ž exists.

We’re told in the question that our function 𝑓 is differentiable at π‘₯ is equal to two. If we let π‘Ž equal to two in our condition for differentiable function, then 𝑓 prime of π‘Ž, which is 𝑓 prime of two, exists. So by satisfying this condition of a differentiability, we can see that option III is true. 𝑓 prime of two exists.

The second thing we know about functions which are differentiable at a point is that if 𝑓 of π‘₯ is differentiable at π‘₯ equal to π‘Ž, then 𝑓 of π‘₯ is continuous at π‘₯ is equal to π‘Ž. And to be continuous at a point, 𝑓 of π‘₯ must satisfy three conditions. One, that 𝑓 exists at the point π‘₯ equal to π‘Ž. Two, that the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ exists. And three, that the limit as π‘₯ approaches π‘Ž of 𝑓 of π‘₯ is equal to 𝑓 of π‘Ž.

Now, we know that our function 𝑓 is differentiable at π‘₯ is equal to two. So it must be continuous at π‘₯ is equal to two. So if we let π‘Ž equal two, then by our first condition for continuity, 𝑓 of π‘Ž exists, so 𝑓 of two exists. This means that our option II is also true.

If we again let π‘Ž equal to two in our second condition for continuity, we know that our function is differentiable at π‘₯ is equal to two and therefore must be continuous at π‘₯ is equal to two. So the second condition for continuity tells us that the limit as π‘₯ approaches two of 𝑓 of π‘₯ exists. And this means that our option I is also satisfied. Using the conditions for differentiability and continuity at a point, we can therefore say that options I, II, and III are all true. This means that the limit as π‘₯ approaches two of 𝑓 of π‘₯ exists, 𝑓 of two exists, and 𝑓 prime at π‘₯ is equal to two also exists.

We can also note that since a function is differentiable at π‘₯ is equal to two, the limit as π‘₯ approaches two of 𝑓 of π‘₯ is equal to 𝑓 of two, which satisfies condition three for continuity. It’s also worth noting that our second point of 𝑓 of π‘₯ is differentiable at π‘₯ is equal to π‘Ž. Then 𝑓 of π‘₯ is continuous at π‘₯ is equal to π‘Ž is not necessarily true in the opposite direction. This means that if a function is continuous at a point, that doesn’t necessarily mean that it’s differentiable at that point.

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