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Sketch the graph of a function $f$ for which $f(0)=0,$ $f^{\prime}(0)=0,$ and $f^{\prime}(x) > 0$ if $x< 0$ or $x > 0$

Calculus 1 / AB

Chapter 2

THE DERIVATIVE

Section 2

The Derivative Function

Functions

Limits

Derivatives

Differentiation

Continuous Functions

Applications of the Derivative

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in this problem. Um, we want to scrap. We want a sketch. A graph of a function where f zero equals zero f prime of zero equals zero f Prime of X is greater than zero for all X, less than zero or X greater than zero. So let's go ahead and draw axes Your wine. I'm sorry, your ex and your wife and we know So we know the point zero comma zero because we have, um we know that f of 00 And we also know that our slope is gonna be zero at that point. So there's, like, a potential tangent line. Well, actually, it would be a change that line. Um, so then what else do we know? So we know that, um, when X is less than zero, we have positive slope, so it's gonna be like going up, and we know that when X is greater than zero slope is also greater than zero. So that means after this point, the slopes gonna be going up. And if the slope, if the if the function is increasing after a 00 then it's gonna be above the X axis. And if it's increasing before 00 Then we know it's below the X axis because it has to come up to meet that 0.0 So from here, you can just pick any function, you know, that satisfies these conditions. So after top my head, like, why equals X cubed? That kind of shape is gonna mimic the behavior that's described in this problem. So we have zero slope in the center and then positive slope on the left and right. Get rid of these rings. We got rid of everything, something like this.

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