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.

2. Intro to Derivatives

Differentiability

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

Verified step by step guidance

To determine if the function f(x) is continuous at x=2, check if the limit of f(x) as x approaches 2 from both sides is equal to f(2).

Examine the graph to see if there is a break, hole, or jump at x=2. If the graph is unbroken and smooth at x=2, the function is continuous there.

To determine if the function is differentiable at x=2, check if the graph has a sharp corner or cusp at x=2. If the graph is smooth and has a well-defined tangent at x=2, the function is differentiable there.

Look at the slope of the graph around x=2. If the slope approaches the same value from both sides as x approaches 2, the function is differentiable at that point.

Based on the graph, if the function is both continuous and smooth (without sharp corners) at x=2, then it is both continuous and differentiable at that point.