Determine whether the following statements are true and give an explanation or counterexample. a. If f'(x) >0 and f'"(x) < 0 on an interval, then f is increasing at a decreasing rate. O A. True. f'(x) > 0 indicates that f is increasing and f''(x) <0 indicates that f' is decreasing. B. False. The rate at which f is increasing cannot be determined. OC. False. f'(x) > 0 indicates that the rate of f is increasing consistently. O D. False. f'"(x) < 0 indicates that f is decreasing and f'(x) > 0 indicates that f''is increasing. b. If f'(c) >0 and f'(c) = 0, then f has a local maximum at c. A. True. These conditions satisfy the second derivative test for a local maximum. B. False. The function f is increasing on an interval containing c and may have an inflection point at c. C. False. The function f has a local minimum at c. D. False. The function f is decreasing on an interval containing c and may have an inflection point at c. c. Two functions that differ by a constant increase and decrease on the same intervals. A. True. The derivative of any constant term is 0, so constant terms do not affect the intervals on which a function increases or decreases. B. False. The function f(x) = x +213 increases everywhere and the function g(x) = x- 337 decreases everywhere. OC. False. The function f(x) =2(x - 1) decreases on (- 0,2) and increases on (2,00) and the function g(x) = 3(x – 1) decreases on (- 00,3) and increases on (3,00). D. False. The critical points of the two functions differ by the same constant. O O O O

Q: Let h(x) = f(g(x)) and p(x) = g(f(x)). Use the table below to compute the following derivatives. a.…

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Q: Below is the graph of f'(x) (the derivative of f(x)). Assume that f(x) is continuous for every real…

A: Consider the given graph: f'2=f4−f04−00=f4−104f4=10

Q: 4. Let g(2) = 3 and g'(2)=1. Find g(-2) and g'(-2) assuming a) g(x) is an even function. b) g(x) is…

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Q: Assume the entire graph of f(x) is shown below. a. Compare f '(1) and f '( 2). Select one. 4+ O A.…

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Q: 10- 10 -10

A: Given function is

Q: Q1. Derive the following formula for approximating the derivative f' and show that it is O(h) by…

A: Given: The formula is f'(x)≈112h -f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h). To prove: The formula and O(h4)…

Q: A particle moves along the x-axis with position function a (t) = 4 sin for 0 <t< 4n where t is…

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Q: Vhich of the following are possible f '(x) for a function f(x) that is decreasing for every x > 5?…

A: We are asked to find which is decreasing

Q: 4. Find the second derivative of each of the following and determine based on your results if the…

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Q: The graph of f' (not f) is given below. At which of the marked values of x is A. f(x) greatest? x =…

A: From the given graph, we have to marked values of x.

Q: 3. Refer to the graph to the right (This represents f'(x)) a. Is f(x) increasing or decreasing at x…

A: According to Bartleby guidelines I can only answer the first three sub-questions of the given…

Q: Use the graph of y =f'(x) to the right to discuss the graph of y = f(x). Organize your conclusions…

A: f'(x) is always below x-axis, hence always negative.Since f'(x) is always negative then f(x) is…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.