Use the derivative f'(x) = (x-4)(x+ 1)(x+ 5) to determine the local maxima and minima of f and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). The local maximum/maxima is/are at x= (Use a comma to separate answers as needed.) The local minimum/minima is/are at x = . (Use a comma to separate answers as needed.) The interval(s) of increase is(are) (Type your answer in interval notation. Use a comma to separate answers as needed.) The interval(s) of decrease is(are) (Type your answer in interval notation. Use a comma to separate answers as needed.) Which is a possible graph of f?

Q: Find the domain and range of f(x)= (ex)/(1+ex)

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Q: use the second derivative test to find the local exterma of the function f(x) =x3 -9x2-48x+72 on R

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Q: determine all values of x at which the function is discontinuous: 1. f(x)=2x/(x^2-1) 2. f(x)=…

A: Given function is  fx=2xx2-1 The function fx is undefined when x2-1=0 which gives x=±1. So, the…

Q: Find the derivative using the formal definition f (x) = 3x2- 4x – 2

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Q: Sketch the graph of f(x)=x² 50x300 and its derivative, on one set of axes, for 10≤x≤ 10. Set…

A: The scale and the viewing window of the graphing utility must be adjusted to obtain a clear graph.

Q: Use Intermediate Vaule Tun to show that the functions f(x) = x^3 and g(x) = 2x + 3 intersect at at…

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Q: Find the function ƒ with the following properties. ƒ'(x) = 3x2 - 1; ƒ(0) = 10

A: Given, f'x=3x2-1f0=10

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.