(a) Sketch the graph of a function that has a local maximum...

(a) Sketch the graph of a function that has a local maximum of $ 2 $ and is differentiable at $ 2 $. (b) Sketch the graph of a function that has a local maximum of $ 2 $ and is continuous but not differentiable at $ 2 $. (c) Sketch the graph of a function that has a local maximum of $ 2 $ and is not continuous at $ 2 $.

(a) Sketch the graph of a function that has a local maximum of $ 2 $ and is differentiable at $ 2 $.
(b) Sketch the graph of a function that has a local maximum of $ 2 $ and is continuous but not differentiable at $ 2 $.
(c) Sketch the graph of a function that has a local maximum of $ 2 $ and is not continuous at $ 2 $.

Key Concepts

Continuity

Continuity of a function at a point means that as the input approaches that point, the output of the function approaches the function’s value at that point. This ensures there are no abrupt jumps or breaks in the graph at that point. Continuity is a fundamental property of functions in calculus and is often a prerequisite for differentiability, though a function can be continuous without being differentiable.

Differentiability

Differentiability refers to the existence of a derivative at a point, meaning that the function has a well-defined tangent at that point and that the limit of the difference quotient exists. This property implies that the function is locally linear around that point and does not have any sharp corners or cusps, which is crucial when analyzing smooth changes in the function's behavior.

Local Maximum

A local maximum is a point on the graph of a function where the function’s value is higher than all other function values in some open interval around that point. In other words, within a small neighborhood of that point, no other value exceeds the value at the local maximum. This concept is key in understanding optimization and the behavior of functions in calculus.

Transcript

00:01 We have three parts in this problem.

00:03 Part a, we sketch the graph of a function that has a local maximum at 2 and is differentiable at 2.

00:12 Part b, we sketch the graph of a function that has local maximum at 2 and is continuous but not differential at that point.

00:23 And in part c, we sketch the graph of a function that has local maximum at 2 and is not continuous at that point.

00:31 So, start with part a, the graph of a function that has a local maximum at 2 and is differentiable at 2.

00:40 Here, by the fermat theorem, we know that the derivative that exists because the function is differentiable at 2 get to be equal to 0 because we have a local maximum.

00:52 So this is a sketch of a graph of a function that has a local maximum at 2 and as we can being differentiable at 2 the graph has a horizontal tangent line at this point of tangency which which of course or where occurs the local maximum so by firmat theorem f derivative at 2 get to be equal to zero that's because we know we are saying that the function has a local maximum at 2 and is differentiable there.

01:46 So then we apply fermatheurin, we know the 3 and get to be 0.

01:50 So this is an example of that.

01:56 We have a local maximum value at.

02:05 Okay, in part b we want a graph of a function that has a local maximum of 2 again, but now the function is continuous but not differentiable at 2.

02:15 And this is an example of that...

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