Based on the graph of f ( x ) f\left(x\right) f ( x ) , describe where the derivative curve f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) would be below the x-axis.

in this problem, we are told, based on the graph, f of x, this function, describe where the derivative curve, f prime of x, would be below the x axis. Now to solve this problem, what we need to do is figure out on our function curve, f of x, where the tangent lines are going to have negative slopes. Because wherever the tangent lines have negative slopes, that's where we're going to be below the x axis on our derivative graph. Now the way that I can do this is I can think about how the tangent lines are going to increase versus decrease. And if I imagine there's, like, a little person that is walking on this hill here, whenever they are going up, the tangent lines are going to be positive. Whenever they are on the top of one of these peaks or these valleys here, that is where the tangent lines are going to be flat. And whenever they're going down the hill, whenever they're going down, that means that the tangent lines are going to be negative. They're gonna have a negative slope. So the thing I need to do is figure out all the places that this little person would go downhill. And I can see that they would go downhill right about there, and they would go downhill right about here. So these are the two places where the tangent lines are going to be negative. And you could try drawing a couple tangent lines to see any place that you have will give you a negative sloped tangent line, and same thing will happen over here. So because of this, we can see that these are the two places or the two intervals where we're going to see the derivative graph be below the x axis. So I need to define what these intervals are. Well, I can see that one place is going to be from negative one. I'm gonna do this here, so we'll have this point negative one all the way over here to 0 on the x axis. So our first interval is going to go from negative one all the way over to 0. Now next, we're going to have the next portion of the graph where we're going downhill, which is going to be from positive one all the way to well, this just keeps on going. And since it keeps going to the right, we would say it goes from positive one all the way to infinity on the x axis. So this right here would be the intervals for which we would be below the x axis on our derivative curve. So that is how you can solve this problem. Hope you found this video helpful.

2. Intro to Derivatives

Basic Graphing of the Derivative

Business Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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Multiple Choice

Based on the graph of $f\left(x\right)$ , describe where the derivative curve $f^{\prime}\left(x\right)$ would be below the x-axis.

(- $\infty$ ,-1) U (0,1)

(-1,0) U (1, $\infty$ )

(- $\infty$ , -1) U (1, $\infty$ )

(-1,1)

Verified step by step guidance

Step 1: Recall that the derivative f′(x) represents the slope of the tangent line to the graph of f(x) at any given point. When f′(x) is below the x-axis, the slope of f(x) is negative, meaning the graph of f(x) is decreasing.

Step 2: Analyze the graph of f(x). Identify the intervals where the graph is decreasing. These intervals occur between turning points where the slope transitions from positive to negative or vice versa.

Step 3: Observe the graph. The function f(x) is decreasing in the intervals (-1, 0) and (1, ∞). In these intervals, the slope of f(x) is negative, so f′(x) would be below the x-axis.

Step 4: Note that the derivative curve f′(x) is not directly visible, but its behavior can be inferred from the graph of f(x). The intervals where f′(x) is below the x-axis correspond to the decreasing portions of f(x).

Step 5: Conclude that the correct intervals where f′(x) is below the x-axis are (-1, 0) U (1, ∞). This matches the behavior of the graph of f(x) as observed.