(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2 . (b

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

Key Concepts

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.

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.

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 So, we are interested in finding example of a function which is continuous at a point and also differentiable, continuous at x equal to 2 and also differentiable at x equal to.

00:13 The best example i can give you is a polynomial.

00:16 F of x is equal to 4x minus x square.

00:19 Let's take this interesting polynomial.

00:21 So if you can draw the graph of this, it is actually a parabola with roots at 0 and 4.

00:27 So it basically looks like this.

00:29 It passes through origin and cuts exactly.

00:32 Is again at 4.

00:33 Now this point at 2 is the point of local maximum.

00:38 How to check the point of local maximum? i think we can use derivative test.

00:41 So when you differentiate the function f f f of x, i'll get f dash of x 4 minus 2 x equated to 0 then x equal to 2 is the critical point.

00:50 Now if you take the double derivative of this it is actually negative.

00:53 So when double derivative is negative the point is local maximum the point x equal to 2 is a point of local maximum.

00:59 And also the function is differentiable at x equal to 2 because i can draw a unique tangent at that point in the neighborhood of that point of course that tangent is parallel to x -axis so this is the best example of a function which is continuous at x is equal to two differentiable at x is equal to 2 and local max at x is equal to at x is equal to right now let's take an example of a function which has local maximum at x equal to 2 it is also continuous but not differentiable so the best example i can give you is i'll just draw the graph first then i'll write the function so this is an example so this is 0 and this is 2 and this is 4 and this is 2 comma 2 so the function i would like to write in this case is it is x when x is less than 2 it is 4 minus x when x is greater than or equal to 2 this is the function this function is discontinuous at x equal to 2 why it is discontinuous because the left hand limit limit extending to 2 minus f of x is basically 2 but the right hand limit limit sorry it is continuous i am so sorry it is continuous.

02:24 So, limit extending to 2 plus f of x is 4 minus 2, that is also 2, and f of 2 2 is also 2.

02:30 Basically, when left hand limit, right hand limit and the value of a function at that point, all the 3 are same, we say that the function is continuous.

02:38 So this function is continuous.

02:40 But is it differentiable? answer is no.

02:43 Because this is a corner.

02:46 Because the graph is not going smoothly at this point.

02:48 It's a corner...

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