Determine if the graph of the function f ( x ) f\left(x\right) f ( x ) is continuous and/or differentiable at x = 1 x=1 x = 1 .

give this problem a try. So here we're asked to determine if the graph of the function f of x is continuous and or differentiable at x equals 1. Okay. Now to solve this problem, I'm first going to figure out whether or not this graph is continuous. So the way that I can do this is by tracing this graph with my pen. Now as I can see, as I trace this graph, I can draw this all in one stroke. I don't have to pick up my pen at any point. So because of this, we would see that this graph is continuous. Now is the graph differentiable? Well, being continuous is one of the criteria for the graph being differentiable, but there's another criteria we have to meet also. We cannot have any sharp corners or turns for a graph to be differentiable at a point. Now looking at this point x equals 1, it appears that there clearly is a sharp corner right there. There's a corner or return that happens sharply, and because of that, that means the graph is not going to be differentiable. So we could see that this function is continuous, but not differentiable, and that would be the solution. So 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 $x=1$ .

Continuous and non-differentiable

Continuous and differentiable

Discontinuous and non-differentiable

Discontinuous and differentiable

Verified step by step guidance

Step 1: To determine if the function f(x) is continuous at x=1, check if the limit of f(x) as x approaches 1 from both sides is equal to f(1).

Step 2: Observe the graph at x=1. The function appears to have no breaks, jumps, or holes at this point, suggesting that the limit from the left (x approaches 1 from the left) and the limit from the right (x approaches 1 from the right) are equal to f(1).

Step 3: To determine if the function is differentiable at x=1, check if the graph has a sharp corner or cusp at this point. Differentiability requires the function to be smooth, without any sharp turns.

Step 4: Examine the graph at x=1. If there is a sharp corner or cusp, the function is not differentiable at this point. If the graph is smooth, then it is differentiable.

Step 5: Based on the observations, conclude whether the function is continuous and/or differentiable at x=1. If the function is continuous but has a sharp corner, it is continuous and non-differentiable at x=1.