Sketch the graph of a differentiable function f that...

Sketch the graph of a differentiable function $f$ that satisfics the given conditions, if possible. If it's not possible, explain how you know it's not possible.
$$f(1)=-1, f^{\prime}(x)<0 \text { for all } x \neq 1, \text { and } f^{\prime}(1)=0$$

Key Concepts

Monotonicity and Strictly Decreasing Functions

A function is strictly decreasing on an interval if for any two points in that interval, a larger input results in a smaller output. This property is closely tied to the sign of the derivative; if f??(x) is negative throughout an interval, the function must decrease without any local flat or increasing behavior.

Differentiability and the Derivative

A function being differentiable means that its derivative exists at every point in its domain and provides the instantaneous rate of change of the function. The derivative, defined as the limit of the difference quotient, reflects how the function behaves in an infinitesimal neighborhood around a point.

Relationship Between the Derivative at a Point and Nearby Behavior

The derivative at a specific point is the limit of the difference quotient as one approaches that point. If every nonzero increment away from the point yields a strictly negative quotient (due to the function's strict decrease), the limit—and hence the derivative—must also be negative. A derivative equal to zero at that point would contradict this limiting behavior, indicating that the combination of a zero derivative at the point and negatives elsewhere is inconsistent.

Transcript

00:01 Next two problems, we're asked to find functions that satisfy certain criteria.

00:08 The first one, we wanted to find a function that is always greater than zero, that is zero, that is one when x is zero, and its derivative is always less than zero.

00:20 So an easy function, i mean, there's more than just, it's a lot, probably an infinite number of functions that satisfy this, but the probably the simplest one is e to the minus x.

00:32 So we can see, we could also have ones that basically, like, ascental to something like this.

00:40 I think there would be, like, our tangent functions, and then also just functions like x squared plus 1 over x squared, say, plus one here, minus one here, stuff like that.

00:54 There were some examples of them previously.

00:56 But this is the easiest one, and we can see that the derivative is minus either the minus f.

01:03 This is always positive.

01:06 When x is 0, it's 1, and this is always negative.

01:09 So we have that, and so we can see here, it's just a function that is continually decreasing, that goes through 0 -1.

01:17 Now, in this case, they wanted us to find a function that was minus 1 when x is 1.

01:23 Its derivative is always 0 for all x except for 1 at 1, and its derivative is 0 when x is 0 when x.

01:34 X is 1.

01:36 So we can basically what we can do is we could use just try to find a polynomial is the best thing to do here.

01:48 And so we want something that's minus 1 when x is 1.

01:53 So we're going to need something that's cubic because we need we need to have something that has something like this.

02:03 So we need at least a cubic...

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