I need you to answer this question and provide me with the explanation for each
ID: 2852411 • Letter: I
Question
I need you to answer this question and provide me with the explanation for each one, please
Below is the graph of the function f(x) from A to H. Use the letters from the graph as the input values where they are. Put the letters on the lines below for each given condition. If the given situation requires an interval, use the letters, (D, F) to represent the interval for input values from D to F. Some conditions require multiple answers and some conditions may not exist. Write NA for those that do not exist. _____________ The derivative of f(x) is positive _________ The derivative of f(x) is negative ______ The derivative of f(x) is 0 _____ The second derivative of f(x) is 0 ________ The derivative of f(x) does not exist ______ There is a relative minimum on f(x) _____ The derivative of f(x) is a maximum _____ The derivative of f(x) is a minimumExplanation / Answer
You MUST know that, graphically, the derivative of a function represents the slope of the tangent line to the function, in a specific point. This is the fundamental part of this concept, in order to understand the exercise. Assume we have a particle, and it’s moving through a road plotted by f(x).
Answer to the question a): between point 1 and point 2 (through CD), we can observe a positive slop of the red dot, because the blue line is increasing (it’s going up from left to right). We have the similar case between point 4 and point 5 (specifically, through FH). It means: (C,D) and (F,H).
Answer to the question b): through AC and DF, the derivative is negative because the slope is decreasing from left to right, which means “it’s going down”. (A,C) and (D,F)
Answer to the question c): if a slope does not increase or decrease, it’ll mean that has a horizontal shape, which means the slope is 0 or, in other words, the derivative of the function, in that specific point, is 0. We have only two points here, and it’s D and F.
Answer to the question d): the second derivative of a function is the derivative of the first one, which means: if you plot the slope of the tangent line (I’ll call it TL1) of the function from point A to point B, you’ll observe the first derivative is non-zero, is equal to a constant value (the slope has a constant value). However, if you plot the slope of the tangent line of TL1, you can realize you’ll be plotting a horizontal line and we know that a horizontal line has a slope of 0. From C to H, the second derivative has a constant value but is not equal to 0. From A until C (C is not included, see below).
Answer to the question e): the derivative of the function, at C, does not exist because there’s something strange: the value of the slope of the tangent line is different if you’re approaching it from the left or from the right. In other words, the slope of the tangent line has several values, so, in conclusion, this point is not differentiable at C.
From (A,C) we don’t have information about the maximum or minimum because the slope doesn’t change at all. There’s a change at D: it means the slope arrived at D, increasing (positive), however, it starts to decrease from D (negative). And there’s another change at F: it means the slope arrived at F, decreasing (negative), and starts to increase (positive)
Maximum: D
Minimum: F
The relative minimum is the lowest point in a particular section of a graph. For our graph, this is point F.