In the given question as follows, (a) find the critical numbers of f, if any, (b) find the open intervals on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema,and (d) use a graphing utility to confirm your results?   f (x) = x/ x - 5

Q: The graph of the derivative of f is shown below: -y-5 4 -3 -2- -1- 0 1 2 f'(x) 3 4 f', the…

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Q: 4. Using the first derivative test to find the location of the local maxima of the following…

A: Given function is  f(x)= x^3 -3x^2 -9x-1

Q: 62²(2³-128) (³+64)³ Consider the function f and its derivatives f(x)=3+64 f'(x)= -2(³-32) (+64)2…

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Q: Let f be a function. Explain, using any method , as to why f(x) and f(x) + 2 have the same…

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Q: Let f(x) = g(x) h(x) and consider the following table of values: The value of f' (-9) is Number…

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Q: Find the relative extrema, if any, of the function. Use the Second Derivative Test if applicable.…

A: The objective is to find relative extrema, if any, of the function.

Q: vertical asymptote(s): horizontal asymptote(s):

A: Since multiple subparts are asked, I am answering first 3 subparts

Q: find the absolute extremum, if any, given by the second derivative test for each function

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Q: Apply your understanding of derivatives to determine any intervals of increase for the function…

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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.