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

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 $x=1$ .

Continuous and non-differentiable

Continuous and differentiable

Discontinuous and non-differentiable

Discontinuous and differentiable

Verified step by step guidance

Step 1: Analyze the graph of the function f(x) at x=1. Observe whether there is a break, jump, or hole in the graph at x=1 to determine continuity. A function is continuous at a point if there is no interruption in the graph at that point.

Step 2: Check if the left-hand limit and right-hand limit of f(x) as x approaches 1 are equal. If they are equal and the function value f(1) exists, the function is continuous at x=1.

Step 3: Examine the slope of the graph at x=1. If the graph has a sharp corner or cusp at x=1, the function is not differentiable at that point. Differentiability requires the graph to be smooth without abrupt changes in direction.

Step 4: Verify if the derivative exists at x=1. A function is differentiable at a point if the derivative can be calculated and is finite. Sharp corners or vertical tangents indicate non-differentiability.

Step 5: Based on the observations, conclude whether the function is continuous and/or differentiable at x=1. Use the definitions of continuity and differentiability to justify your conclusion.