Determine if the graph of the function f ( x ) f\left(x\right) f ( x ) is continuous and/or differentiable at x = 2 x=2 x = 2 .

Let's try solving this problem. So here we're asked to determine if the graph of the function f of x is continuous and or differentiable at x equals 2. So the point x equals 2 is going to be right there, and the way that we can figure this out is by first figuring out if it's continuous and then figuring out if it's differentiable. Because recall that it's possible for a graph to be continuous but not differentiable, but it's not possible for a graph to be differentiable but not continuous at a certain point. So let's see what happens. Well, to figure out if this is continuous, I'm gonna trace this with my pen. And as I can see, I don't actually have to pick up my pen at any point. I can trace this all in one curve, so that means that it is continuous. But the question is, is it now differentiable? Well, if it's continuous, that's one of the criteria for being differentiable. But to figure out if this is differentiable, it can't have any sharp corners or turns. Well, looking at this, I don't see any sharp corners or turns. It looks like it's actually a smooth turn right there. So because of this, we could say it's continuous and differentiable at x equals 2. So we could say that this is continuous and differentiable, and that would be the solution to this problem. So that is how you can solve these types of problems. Hope you found this video helpful.

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2. Intro to Derivatives

Differentiability

Business Calculus

2. Intro to Derivatives

Differentiability

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Multiple Choice

Determine if the graph of the function $f$\left$(x$\right$)$ is continuous and/or differentiable at $x=2$ .

Continuous and non-differentiable

Continuous and differentiable

Discontinuous and non-differentiable

Discontinuous and differentiable

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Step 1: Analyze the graph of the function f(x) at x=2. Observe whether there is any break, jump, or hole in the graph at x=2. If the graph is unbroken and continuous at x=2, the function is continuous at this point.

Step 2: Check the slope of the graph at x=2. If the graph has a smooth curve without any sharp corners or vertical tangents at x=2, the function is differentiable at this point.

Step 3: Recall that for a function to be differentiable at a point, it must also be continuous at that point. Therefore, if the graph is continuous and smooth at x=2, the function is both continuous and differentiable.

Step 4: Based on the graph provided, confirm that the function f(x) does not have any discontinuities (breaks, jumps, or holes) at x=2 and that the slope of the graph changes smoothly without sharp corners or vertical tangents.

Step 5: Conclude that the function f(x) is continuous and differentiable at x=2 based on the observations from the graph.